# A Deep Redshift Survey of the Perseus Cluster: Spatial Distribution and
Kinematics of Galaxies

Wooseok Kang  Department of Physics and Astronomy, Seoul National University,
1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea  Ho Seong Hwang
Department of Physics and Astronomy, Seoul National University, 1 Gwanak-ro,
Gwanak-gu, Seoul 08826, Republic of Korea  SNU Astronomy Research Center,
Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of
Korea  Hyunmi Song  Department of Astronomy and Space Science, Chungnam
National University, Daejeon 34134, Republic of Korea  Changbom Park  School
of Physics, Korea Institute for Advanced Study (KIAS), 85 Hoegiro, Dongdaemun-
gu, Seoul, 02455, Republic of Korea  Narae Hwang Korea Astronomy and Space
Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Korea
Byeong-Gon Park  Korea Astronomy and Space Science Institute, 776 Daedeokdae-
ro, Yuseong-gu, Daejeon 34055, Korea

###### Abstract

We study the global kinematics of the Perseus galaxy cluster (Abell 426) at
redshift z = 0.017 using a large sample of galaxies from our new MMT/Hectospec
spectroscopic observation for this cluster. The sample includes 1447 galaxies
with measured redshifts within 60′ from the cluster center (1148 from this
MMT/Hectospec program and 299 from the literature). The resulting
spectroscopic completeness is 67% at r-band apparent magnitude
$r_{\mathrm{Petro,0}}\leq 18.0$ within 60′ from the cluster center. To
identify cluster member galaxies in this sample, we develop a new open-source
Python package, CausticSNUpy. This code implements the algorithm of the
caustic technique and yields 418 member galaxies within 60′ of the cluster. We
study the cluster using this sample of member galaxies. The cluster shows no
significant signal of global rotation. A statistical test shows that the
cluster does not have a noticeable substructure within 30′. We find two
central regions where the X-ray emitting intracluster medium and galaxies show
significant velocity differences ($>7\sigma$). On a large scale, however, the
overall morphology and kinematics between the intracluster medium and galaxies
agree well. Our results suggest that the Perseus cluster is a relaxed system
and has not experienced a recent merger.

Intracluster medium (858); Perseus Cluster (1214); Redshift surveys (1378);
Galaxy clusters (584); Catalogs (205)

##  1 Introduction

Clusters of galaxies form through the collapse of initial density fluctuations
and hierarchical merging. Clusters are regarded as laboratories for
astrophysical phenomena, such as high-energy physics and the evolution of
galaxies (Allen et al., 2011). Thus, studies of galaxy clusters and their
components give insight into both cosmology and astrophysics. The constituents
of galaxy clusters are dark matter, galaxies, and intracluster medium (ICM).
The mass fractions of the three components are 80–95% for dark matter, 5–20%
for ICM, and 0.5–3% for galaxies (Ettori et al., 2009).

Different mechanisms are applied to each component. Dark matter and the
stellar component of galaxies are considered collisionless, even though
sometimes interactions between galaxies play a key role in galaxy evolution
(e.g., Park & Hwang, 2009; Hwang et al., 2012). The ICM gas is the collisional
component of the cluster, governed not only by gravity but also by
magnetohydrodynamics and thermodynamics. Because of this difference, dark
matter and galaxies have relaxation timescales different from the gas after a
cluster experiences a merger (Roettiger & Flores, 2000; Clowe et al., 2006;
Shin et al., 2022). Thus, a comparison of the kinematics and spatial
distribution of galaxies and ICM can provide an important hint for the
formation and evolution history of galaxy clusters (e.g., Song et al., 2017,
2018).

Among many clusters that we can study both gas and galaxies, we focus on the
Perseus cluster (Abell 426; A426) in this study which has abundant X-ray data
sets. This cluster is one of the nearest clusters ($z=0.017$) and is the
brightest cluster in the X-ray sky. X-ray is one way to study ICM, as it has
high temperatures ($\sim$keV) and emits X-ray via bremsstrahlung. Previous
research on the ICM of the Perseus cluster focused on the structures and
dynamics of gas in the central region. Fabian et al. (2000) investigated the
structures in the central region of the Perseus cluster with the Chandra data.
Simionescu et al. (2012) compared the X-ray images from Chandra, XMM-Newton,
and Suzaku with numerical simulations and noted that the spiral features of
the gas may be related to a past minor merger. Tamura et al. (2014) measured
the bulk motion of ICM in the core region ($R\lesssim 5\arcmin$) with Suzaku.
They found a velocity structure west of the cluster, where an excess of the
X-ray intensity exists. Hitomi Collaboration et al. (2016) presented high-
resolution spectroscopic measurements of the gas bulk velocity and velocity
dispersion in the central region ($\sim$100kpc) of the Perseus cluster.
Sanders et al. (2020) analyzed the photometric and spectroscopic data from the
XMM-Newton archive. They found a correlation between the ICM velocity and
temperature, especially giving attention to the bulk motion of gas in the
eastern region of the cluster core. The X-ray data of the Perseus cluster have
also been used for the search for dark matter (Tamura et al., 2015, 2019).

On the other hand, there are also active studies on the galaxy population of
the Perseus cluster. For example, Aguerri et al. (2020) investigated the
luminosity function of galaxies and dwarf galaxy populations with 963 newly
measured redshifts. Meusinger et al. (2020) presented a catalog of galaxies in
the Perseus region and analyzed galaxy properties such as active galaxies and
morphology. Khanday et al. (2022) used the Sloan Digital Sky Survey (SDSS;
York et al., 2000) data to examine the scaling relations of galaxies (e.g.,
the fundamental plane). However, there have been few studies focusing on the
combined view of the galaxies and ICM in the Perseus cluster (e.g., Tamura et
al., 2014). This is mainly because of the lack of deep spectroscopic data for
the galaxies in the central region where we can make direct comparisons
between the two components. We have therefore conducted a deep spectroscopic
survey for this cluster and made the comparison between galaxies and ICM.

The aim of this paper is three-fold. First, we present a catalog of galaxies
with measured redshifts in the central region of the Perseus cluster. Second,
we introduce CausticSNUpy111https://github.com/woodykang/CausticSNUpy, a
Python implementation of the caustic technique for cluster membership
determination. Third, we analyze the physical characteristics of the Perseus
cluster, especially by comparing the spatial distribution and kinematics of
galaxies with the X-ray-emitting ICM.

In Section 2, we describe the data used in this study. We elaborate on the
membership determination and examine the physical properties of the Perseus
cluster in Section 3. We discuss and conclude our study in Section 4.
Throughout the paper, we adopt the standard $\Lambda$ cold dark matter
($\Lambda$CDM) cosmology with $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, and
$H_{0}=100h\ \mathrm{km\ s^{-1}\ Mpc^{-1}}$.

##  2 Data

###  2.1 Photometric Data

Figure 1: The size-magnitude diagram of objects in the SDSS photometric
catalog. Gray dots represent point sources identified by SDSS, and green
triangles are objects originally labeled as extended sources but visually
confirmed to be point sources or artifacts. Blue circles and red squares are
galaxies without and with spectroscopic redshifts. Only 20% of the data points
except the green triangles are plotted for clarity.

For the photometric data, we used the SDSS Data Release 17 (DR17; Abdurro’uf
et al., 2022). From this photometric catalog, we selected target galaxies for
observation with MMT/Hectospec. Our goal for the observation was to obtain
complete and uniform spectroscopic data in the Perseus cluster region.
Therefore, we prioritized galaxies according to their r-band apparent
magnitude without any other criteria such as color selection. The SDSS
photometric catalog provides a flag (p_probpsf) that indicates if the object
is likely to be a point source. However, some objects labeled as extended
sources turned out to be images of saturated stars or artifacts (e.g.,
spikes). To exclude such objects from the target list, we conducted a visual
inspection of all the targets plus any objects brighter than 17 mag in
$r$-band were also inspected. A total of 745 of 3331 objects were relabeled as
a result. In Figure 1, we plot the size-magnitude plot of the objects in the
SDSS photometric catalog. We use the Petrosian radius $R_{\rm{Petro}}$ for the
size and the extinction-corrected $r$-band magnitude $r_{\rm{Petro},0}$ for
the magnitude. The green triangles are objects previously flagged as extended
sources, but excluded after visual inspection. The plot confirms that the
excluded objects are not likely to be extended objects.

###  2.2 Spectroscopic Data

Table 1: Observation with MMT/Hectospec Field ID | R.A. | Decl. | Observation Date | Exposure | Number of | Number of
---|---|---|---|---|---|---
| (deg, J2000) | (deg, J2000) |  | (minutes) | Targets | Measured Redshifts
a426c_2 | 49.965949 | 41.511704 | Nov 19 2014 | $3\times 20$ | 258 | 232
a426c21_2 | 49.545053 | 41.179996 | Dec 7 2021 | $3\times 20$ | 259 | 214
a426c21_1 | 50.452605 | 41.179409 | Dec 8 2021 | $3\times 20$ | 258 | 246
a426c21_3 | 49.573721 | 41.940983 | Dec 9 2021 | $3\times 20$ | 254 | 233
a426c21_4 | 50.426230 | 41.941936 | Dec 9 2021 | $3\times 20$ | 254 | 235
Figure 2: Observation fields of MMT/Hectospec in the Perseus cluster region.
Dashed circles indicate the five fields and open circles represent the
positions of the observed galaxies. The red cross shows the center of the
cluster. The Digitized Sky Survey image is shown in the background.  Table 2:
Catalog of objects with redshift measured within 60′ from the Perseus cluster
center

ID | SDSS DR17 ObjIDaaObjIDs of objects not found in DR17 are DR7 ObjIDs. | R.A. | Decl. | $r_{\rm{Petro,0}}$ | Flagbb(0) Extended source, (1) point source | zccIf redshift errors of galaxies from NED are not available, we set them to 0.0001. | z Sourcedd (1) This study, (2) SDSS, (3) Bilicki et al. (2014), (4) van den Bosch et al. (2015), (5) Brunzendorf & Meusinger (1999), (6) Falco et al. (1999) (7) Gandhi et al. (2004), (8) Huchra et al. (1999), (9) Jørgensen et al. (2018), (10), Miller & Owen (2001), (11) Paturel et al. (2003), (12) Penny & Conselice (2008), (13) Penny et al. (2014), (14) Poulain et al. (1992), (15) Smith et al. (2000), (16) Springob et al. (2005), (17) Zaw et al. (2019), (18) Meusinger et al. (2020) | Memberee(0) Non-member, (1) Member of the Perseus cluster
---|---|---|---|---|---|---|---|---
|  | (deg, J2000) | (deg, J2000) | (mag) |  |  |  |
1 | 1237661121850770550 | 48.627453 | 41.415013 | 18.485 | 1 | $-0.000259\pm 0.000012$ | 2 | 0
2 | 1237661083198882927 | 48.632691 | 41.555350 | 18.478 | 1 | $-0.000057\pm 0.000011$ | 2 | 0
3 | 1237661121850835912 | 48.646550 | 41.309490 | 20.069 | 1 | $-0.000717\pm 0.000037$ | 2 | 0
4 | 1237661121850835851 | 48.656534 | 41.362330 | 18.247 | 1 | $-0.000049\pm 0.000017$ | 2 | 0
5 | 1237661083198882258 | 48.657297 | 41.532471 | 15.817 | 1 | $-0.000272\pm 0.000008$ | 2 | 0
6 | 1237661121850835214 | 48.657786 | 41.343411 | 16.721 | 0 | $-0.000228\pm 0.000010$ | 2 | 0
7 | 1237661121850835724 | 48.669263 | 41.436923 | 16.166 | 0 | $0.134324\pm 0.000016$ | 2 | 0
8 | 1237661121850835439 | 48.681708 | 41.275673 | 17.159 | 1 | $-0.000257\pm 0.000011$ | 2 | 0
9 | 1237661083198882246 | 48.683231 | 41.565523 | 17.622 | 0 | $-0.000038\pm 0.000013$ | 2 | 0
10 | 1237661055281397988 | 48.683388 | 41.823938 | 16.483 | 0 | $-0.000091\pm 0.000009$ | 2 | 0

Note. — This table is available in its entirety in machine-readable form.

We observed 1283 galaxies in the central region of the Perseus cluster with
Hectospec on MMT 6.5m telescope (Fabricant et al., 2005), as summarized in
Table 1. Hectospec is a multiobject spectrograph with 300 fibers whose field
of view is 1° in diameter. We observed a total of five fields with
MMT/Hectospec, covering a region of approximately 1° in radius. The observed
fields are shown in Figure 2. After the observation, we reduced the data with
HSRED v2.0, an IDL pipeline provided by the MMT Observatory. With the
resulting 1D spectra, we measured the redshifts of the galaxies using RVSNUpy
(Kim et al. in preparation). RVSNUpy is a Python package that measures the
redshift of a spectrum by cross-correlating the given spectrum with various
spectral templates. Along with the redshift value for each spectrum, the
package yields the Tonry & Davis (1979) $r$ value ($r_{TD}$) which can be used
as a measure of reliability. Geller et al. (2014b, 2016) confirmed by visual
inspection that redshifts with $r_{TD}>4$ are reliable. We used galaxies whose
$r_{TD}$ are greater than 4.5 to be conservative. The sum of the number of
reliable redshifts in each field is 1160, as can be inferred from Table 1.
Among the galaxies with reliable redshifts, 9 were observed twice and have two
redshift values. For these galaxies, we use the redshifts with higher
$r_{TD}$. This results in 1151 galaxies with reliable redshifts, among which
1148 galaxies are within 60′ from the center of the cluster. We also
incorporated redshift data from the literature. We retrieved 502 redshifts
from SDSS DR17, 35 from NASA/IPAC Extragalactic Database (NED), and 9 from
Meusinger et al. (2020); among those, 265 galaxies (SDSS), 29 galaxies (NED),
and 5 galaxies (Meusinger et al., 2020) are within 60′. Thus, the total number
of galaxies with measured redshifts at $R<60\arcmin$ is 1447. It should be
noted that there was an extensive galaxy redshift survey on this cluster by
Aguerri et al. (2020) who measured 963 redshifts of the galaxies. However, we
could not include them in this study because their data are not publicly
available. We omit them in the evaluation of the usefulness of our
MMT/Hectospec data in the following paragraphs. We note that they covered the
cluster region with $R<147\arcmin$. However, there is only one pointing out of
21 pointings covering $R<30\arcmin$, suggesting that the overlap of the data
between the two studies is not significant.

Figure 3:  Comparison of redshifts between sources (left) and within
MMT/Hectospec observation in this study (right). On the left panels,
$z_{\rm{MMT}}$ and $z_{\rm{Lit}}$ denote redshifts measured by MMT/Hectospec
and those from the literature, respectively. On the right panels, $z_{2}$ are
redshifts with higher $r_{TD}$ than $z_{1}$.

We compare the redshift measurements of galaxies in Figure 3. The left panels
show the comparison between redshifts in the literature ($z_{\rm{Lit}}$) and
the redshift measured by MMT/Hectospec in this study ($z_{\rm{MMT}}$). There
are three overlapping galaxies from SDSS (red circle) and one from Meusinger
et al. (2020) (blue star). The plot shows a general agreement between the two.
The right panels show the comparison of redshifts of the 9 galaxies measured
twice with MMT/Hectospec in this study. The plot again shows that the
repetitive measurements agree well. We denote $z_{2}$ as the redshift with a
higher $r_{TD}$ and $z_{1}$ as the other.

We constructed a catalog containing the SDSS ObjID, coordinates, extinction-
corrected r-band Petrosian magnitudes $r_{\rm{Petro},0}$, measured redshifts,
and cluster membership (see Section 3.1 for membership determination). The
catalog is given in Table 2. For the full presentation of the data, we include
in the catalog three galaxies whose spectra are measured by MMT/Hectospec but
outside 60′ from the cluster center.

Figure 4: Color-magnitude diagram of galaxies within 60′ from the cluster
center. Gray dot, green square, red circle, and blue cross indicate galaxies
without spectroscopic redshifts, galaxies with spectroscopic redshifts but
interlopers, red members of the cluster, and blue members of the cluster,
respectively. The black solid line represents the red sequence calculated by
linear fitting. The blue dashed line is $3\sigma$ bluer than the red sequence,
which is the criterion for classifying blue and red members.

Figure 4 shows the color-magnitude diagram (CMD) of galaxies within 60′ from
the cluster center. The cluster members form a clear red sequence in the
diagram. To partition the cluster members into blue and red members, we follow
the method in Hwang et al. (2014). We first draw the red sequence by linear
fitting member galaxies with $r_{\rm{Petro},0}<18$ and
$g_{\rm{model},0}-r_{\rm{model},0}>0.5$. The best-fit line (black solid line)
is $g_{\rm{model},0}-r_{\rm{model},0}=-0.033r_{\rm{Petro},0}+1.26$ with
scatter $\sigma=0.058$. Galaxies bluer than $3\sigma$ from the red sequence
are defined as blue members, and the others as red members. The $3\sigma$
boundary is shown in the blue dashed line in Figure 4. Out of 418 member
galaxies within 60′, 370 are classified as red members and 48 as blue members.
While inspecting the CMD, we found that four cluster members had unusual
apparent magnitudes on the CMD. Those objects have adjacent foreground stars,
which probably affected the magnitude measurements. For the four galaxies, we
replaced the Petrosian magnitudes with the fiber magnitudes.

Figure 5 shows the comparison of the spectroscopic completeness before this
study and with the new MMT/Hectospec data added. We plot the spectroscopic
completeness for two regions: $R<60^{\prime}$ (left) and $R<30^{\prime}$
(right). The top panels show the histograms of all galaxies (black dashed
lines), galaxies with spectroscopic redshifts before this study (solid black
lines), and galaxies with redshifts obtained in this study (solid red lines),
as functions of $r_{\rm{Petro},0}$. The bottom panels show the differential
completeness as functions of $r_{\rm{Petro},0}$ before this study (solid black
lines) and with data used in this study (solid red lines). The black and red
vertical dotted lines indicate the magnitudes at which the differential
completeness drops below 50%. The 50% magnitude limit was pushed from 16.1 mag
to 18.0 mag for galaxies within 60′, and from 16.1 mag to 19.1 mag for those
within 30′. The cumulative spectroscopic completeness is 67% for galaxies with
$R<60\arcmin$ and $r_{\mathrm{Petro,0}}\leq 18.0$, and 76% for $R<30\arcmin$
and $r_{\mathrm{Petro,0}}\leq 19.1$. As a result, our new data play an
essential role in increasing the spectroscopic completeness, especially for
the $R<30\arcmin$ region.

Figure 5: Top left: histogram of galaxies within 60′ from cluster center for
all galaxies (black dashed line), galaxies with redshifts before this study
(solid black line), and galaxies with redshifts from this study (solid red
line). Bottom left: differential spectroscopic completeness as a function of
magnitude for data before this study (solid black line) and from this study
(solid red line). Black and red dotted vertical lines show the magnitudes
corresponding to the differential completeness of 50%. The magnitudes are 16.1
mag and 18.0 mag, respectively. Top right and bottom right: Same as the left
panels, but for galaxies within 30′. The magnitudes at which the completeness
is 50% are 16.1 mag for data before this study and 19.1 mag for data from this
study.

To confirm the spatial uniformity of the completeness, Figure 6 shows the 2D
spectroscopic completeness. The fields of view for (a) and (b) of Figure 6 are
$120\arcmin\times 120\arcmin$ and $60\arcmin\times 60\arcmin$, corresponding
to $R<60\arcmin$ and $R<30\arcmin$, respectively. The bottom left panels of
(a) and (b) show the 2D spectroscopic completeness with the centers of the
field of view aligned to the cluster center. The top panels and the right
panels show the marginal completeness along R.A. and Decl., respectively. The
completeness is calculated for the galaxies brighter than 18.0 mag for (a) and
19.1 mag for (b), which are the 50% magnitude limit as described in the
preceding paragraph. The completeness for $R<60\arcmin$ is high in the central
region but drops rapidly at the edges. On the other hand, albeit with the
fainter magnitude limit, the $R<30^{\prime}$ region maintains overall uniform
and high completeness. The reason for the higher completeness in the inner
region is due to the overlapping observation fields in the central region as
shown in Figure 2. This is consistent with the results from Figure 5. For this
reason, we use the $R<30\arcmin$ region with a magnitude limit of 19.1 for
analyses where spatial distribution is considered important.

Figure 6: (a) Bottom left: Spectroscopic completeness of galaxies with
$r_{\rm{Petro},0}\leq 18.0$ as a function of right ascension and declination.
The field size is $120\arcmin\times 120\arcmin$ with its center aligned to the
center of the Perseus cluster. Marginal completeness in the right ascension
(top) and declination (right) are also given. (b) Same as (a), but for the
galaxies brighter than 19.1 mag in the $60\arcmin\times 60\arcmin$ field.
Figure 7: (Top) Distribution of $r_{\rm{Petro},0}$ along redshift. Black dots
represent individual galaxy redshifts before this study, while red dots
represent galaxies observed with MMT/Hectospec. (Bottom) Black and red lines
show the histogram of the whole galaxies with spectroscopic redshifts and
galaxies whose redshifts are measured with MMT/Hectospec in the $R<60\arcmin$
region, respectively. The blue histogram shows the galaxies determined as
cluster members.

Figure 7 shows the comparison between the redshift distribution of the data
available before this study and the data used in this study. The top panel
shows the apparent magnitudes of galaxies as a function of redshift. Black
dots indicate galaxy data before this study, and red dots indicate those
observed with MMT/Hectospec. The bottom panel shows the number of galaxies
with measured redshifts. The black line represents all galaxies, consisting of
those from MMT/Hectospec, SDSS, and NED. The red hatched histogram shows only
the galaxies observed with MMT/Hectospec. The blue histogram represents the
member galaxies of the Perseus cluster. As expected, the member galaxies are
concentrated around the redshift of the cluster center ($z=0.01756$). As
Figure 7 proves, the MMT/Hectospec data complement the existing data set with
faint galaxies of relatively higher redshifts.

##  3 Results

###  3.1 Cluster Membership Determination

To analyze the cluster property, the separation of cluster member galaxies
from background and foreground galaxies is necessary. One method to determine
cluster membership is to plot galaxies in the 2D pseudo phase space known as
the redshift space (radial velocity v.s. projected clustercentric distance;
middle left panel of Figure 8). Galaxies in a cluster form a trumpet-shaped
distribution in the redshift space, whose envelopes are referred to as
caustics (Kaiser, 1987; Regos & Geller, 1989; Diaferio & Geller, 1997). The
caustics are related to the escape velocity at a given radius and in turn,
galaxies lying outside the caustics can be regarded as interlopers (Diaferio,
1999). Serra & Diaferio (2013) tested the caustic technique on galaxy clusters
produced from N-body simulation clusters and reported that 95% of the true
cluster members were recovered and only 8% of the interlopers were identified
as members within $3r_{200}$, where $r_{200}$ is the radius at which the
average density within a sphere is equal to 200 times the critical density of
the universe. This confirms that the caustic technique is reliable for
distinguishing cluster members from the back- and foreground galaxies.

Figure 8:  The caustics are plotted in the middle left panel, along with red
member galaxies (red circles), blue member galaxies (blue crosses), and non-
member galaxies (black open circles). The bottom left panel shows the
distribution of apparent magnitude as a function of clustercentric distance.
Histograms of galaxies in the field of the Perseus cluster are represented
with respect to clustercentric distance (top left), radial velocity (middle
right), and apparent magnitude (bottom right). Red and blue histograms are for
red and blue members, respectively. Black histograms are for all member
galaxies, while open histograms are for all observed galaxies.

There are software implementations of the caustic technique. For example, Ana
Laura Serra & Antonaldo Diaferio (2014) developed the Caustic App, a
standalone software tool for applying the caustic method. The software was
used in previous studies such as Song et al. (2017, 2018). Their tool is
limited to a certain operating system (Linux) and does not support recent OS
versions. On the other hand, a Python package called causticpy is also
available. The package is based on Gifford et al. (2013) and Gifford & Miller
(2013). Different assumptions and implementation details are adopted in
Gifford & Miller (2013) than in the traditional caustic technique, including
the assumption of Navarro-Frenk-White profile and using static kernel density
estimation.

In this study, we develop CausticSNUpy, a Python code to run the caustic
technique. The code follows the procedures elaborated in detail by Diaferio
(1999), Serra et al. (2011), and Serra & Diaferio (2013). The code first finds
candidate members using a hierarchical clustering algorithm. The code then
calculates the galaxy number density in the redshift space via adaptive kernel
density estimation. The caustics are where the number density reaches a
certain threshold. The final member galaxies are determined as those lying
inside the caustics in the redshift space. In Appendix A, we compare the
caustics of our code with that of the Caustic App.

The caustic method also provides an estimate of the cluster center. The right
ascension and declination of the center of the cluster are calculated as the
peak of the galaxy number density on the sky, and the redshift as the median
of the candidate member redshifts (Serra et al., 2011). The center of the
Perseus cluster calculated by the caustic method is $(03^{\rm h}19^{\rm
m}21\fs 70,41\arcdeg 29\arcmin 42\farcs 44)$ and $cz=5006\mathrm{km\ s^{-1}}$.
The brightest cluster galaxy (BCG) can also be used as the cluster center. For
the Perseus cluster, the BGC is NGC 1275 whose coordinates are $(03^{\rm
h}19^{\rm m}48\fs 16,41\arcdeg 30\arcmin 42\farcs 1)$ and $cz=5264\mathrm{km\
s^{-1}}$ (Hudson et al., 2001). The offset between the cluster center derived
from the caustic technique and NGC 1275 is 5$\farcm$05 on the sky and
$258\rm{km\ s^{-1}}$ in redshift. We fix the cluster center to the coordinates
of NGC 1275 for consistency with other studies.

The middle left panel of Figure 8 shows the result of the caustic technique
run with CausticSNUpy. The black solid line represents the caustics. The red
circles, blue crosses, and black open circles indicate red members, blue
members, and non-members, respectively. A total of 468 galaxies were
determined as cluster members out of 1699 galaxies with spectroscopic
redshifts. Among them, 418 galaxies were identified as members within 60′ and
246 galaxies within 30′.

The top left and middle right panels show the histograms of galaxies along the
clustercentric distance and the radial velocity. Black, red, and blue hatched
histograms represent the histograms of all members, red members, and blue
members, respectively. The empty histogram is for all galaxies with redshifts.
The distribution of apparent magnitudes is shown in the bottom left panel. The
magnitude limits for MMT/Hectospec (20.5 mag) and SDSS (17.7 mag) are apparent
in $R<60\arcmin$ and $R<90\arcmin$, respectively.

###  3.2 Global Kinematics of the Cluster Traced by Galaxies

####  3.2.1 Velocity Dispersion Profile

Figure 9:  Radial velocity dispersion profile of galaxies. On both panels, the
black circles represent the velocity dispersion of all member galaxies. (Top)
Galaxies are divided into red member galaxies (red open circles) and blue
member galaxies (blue crosses). (Bottom) Galaxies are divided by their
absolute magnitude. Magenta squares represent bright member galaxies and cyan
plus signs represent faint member galaxies. Only bins with three or more
galaxies are used. Error bars of galaxy velocity dispersion are resampling
errors.

Here, we present the velocity dispersion profile of member galaxies in the
Perseus cluster (Figure 9) and provide related scientific questions. In Figure
9, we divide the sample into subsets according to two criteria: color (top)
and absolute magnitude (bottom). This is to investigate the dependence of
velocity dispersion profile on cluster mass tracer properties. The colors of
member galaxies are defined in Section 2.2, and the absolute magnitude is
calculated by taking into account the K- and E-correction,

$M_{r}=r_{Petro,0}-DM-K(z)+E(z)$ (1)

where $M_{r}$ is the $r$-band absolute magnitude, $DM=5\log(D_{L}/10\rm{pc})$
is the distance modulus, $D_{L}$ is the luminosity distance, $K(z)$ is the
K-correction, and $E(z)$ is the luminosity evolution correction. For the
K-correction, we use the equation presented by Choi et al. (2007):

$\begin{split}K(z)=&3.0084(z-0.1)^{2}+1.0543(z-0.1)\\\
&-2.5\log(1+0.1)\end{split}$ (2)

We calculate the luminosity evolution correction as $E(z)=1.6(z-1)$, given by
Tegmark et al. (2004). We choose $-20$ mag as the magnitude cut to divide
galaxies, which is approximately the characteristic magnitude of a cluster
luminosity function. As the absolute magnitude can used as a proxy for the
stellar mass of a galaxy, we can regard brighter galaxies as more massive
ones.

Previous studies showed that red member galaxies are reliable tracers of the
cluster mass (Rines et al., 2013; Geller et al., 2014a; Song et al., 2017).
Also, in a galaxy cluster, less massive galaxies have joined the system rather
recently. Thus, low-mass member galaxies may show a velocity dispersion larger
than that of the high-mass galaxies because of high velocity anisotropy
(Biviano & Katgert, 2004; Hwang & Lee, 2008). The top panel shows that the
inclusion of blue members does not significantly affect the measured velocity
dispersion profile. In the bottom panel, the velocity dispersion of bright
galaxies shows a slight offset from the total profile due to the small number
of galaxies satisfying $M_{r}\leq-20$.

The velocity dispersion profile of cluster galaxies can provide useful
information about the mass profile of the cluster or the velocity anisotropy
of cluster galaxies. Fixing one of the two parameters allows the estimation of
the other. For example, Hwang & Lee (2008) constrained the velocity
anisotropies of 10 clusters from the velocity dispersion profiles of galaxies
and the mass profiles inferred from X-ray observation data (see also Biviano &
Katgert, 2004). On the other hand, studies such as Katgert et al. (2004)
assumed the velocity anisotropies to calculate the mass profiles of clusters.
Furthermore, Geller et al. (2014a) suggested that the stellar velocity
dispersion profile of the BCG could provide a complementary measure of the
cluster potential. A similar analysis can be done for this cluster with the
dispersion profile in this study, which we leave for future works.

####  3.2.2 Detecting Signals of Rotation

Figure 10:  Rotation velocity of the Perseus cluster. (Top left) Individual
circles indicate the radial velocities of member galaxies within 27′ as a
function of position angle $\theta$. The black line represents the fitted
curve according to Equation 3. (Top right) Difference between the mean
velocities $\langle v_{1}\rangle$, $\langle v_{2}\rangle$ of member galaxies
within 27′ in two regions divided by a line with position angle $\theta$.
(Bottom left and right) Same as the top panels, for galaxies in
$27\arcmin<R<60\arcmin$.  Figure 11:  Rotation profile of the Perseus cluster.
(Top) Rotation amplitude. (Bottom) Position angle of the rotation axis. Values
in blue triangles are measured with Hwang & Lee (2007) method and those in red
circles with Manolopoulou & Plionis (2017). Error bars are resampling errors.
The border between the inner region and the outer region is set to 27′ so that
there is almost the same number of galaxies in each region.

A cluster can be rotating, by obtaining angular momentum from mergers with
other clusters or due to lasting initial angular momentum since the formation
of the cluster. There were several methods to detect and quantify the global
rotation of a cluster. For example, Hwang & Lee (2007) fit the radial
velocities of galaxies with a sinusoidal function with regard to the position
angle on the sky. The fitting function is

$v_{p}(\theta)=v_{sys}+v_{rot}\sin(\theta-\theta_{rot})$ (3)

where $\theta$ is the position angle of each galaxy measured from north to
east, $\theta_{rot}$ is the projected rotation axis on the sky, $v_{p}$ is the
observed radial velocity, $v_{sys}$ is the systematic velocity of the cluster,
and $v_{rot}$ is the rotation amplitude. The observed quantities are $v_{p}$
and $\theta$ and $v_{sys}$, $v_{rot}$, and $\theta_{rot}$ are fitting
parameters. On the other hand, Manolopoulou & Plionis (2017) proposed another
idea. Their algorithm first partitions the galaxies into two regions divided
by a line projected on the sky and passing through the cluster center. The
mean velocities of galaxies in the two regions, $\langle v_{1}\rangle$ and
$\langle v_{2}\rangle$, are calculated. Then, the algorithm rotates the
divider from north to east and finds when the difference between the two mean
velocities $v_{dif}=\langle v_{1}\rangle-\langle v_{2}\rangle$ reaches its
maximum value. The rotated angle of the divider when $v_{dif}$ becomes maximum
is the rotation axis of the cluster center projected on the sky, and
$v_{rot}=\max\\{v_{dif}\\}$ is the rotation amplitude.

In this study, we apply both methods separately to the Perseus cluster. Also,
we divide the data into two groups: member galaxies in $R<27\arcmin$ and in
$27\arcmin<R<60\arcmin$. We choose 27′ to make the number of galaxies in each
group about the same. For this analysis, we assume that the 2D completeness of
galaxies does not play a crucial role and we decide to use all galaxies
regardless of their brightness. The main justification for this assumption is
that the targets observed by MMT/Hectospec were selected only by their
apparent magnitudes, and thus our catalog is unbiased when studying global
kinematics. Figure 10 shows the result of applying the Hwang & Lee method and
the Manolopoulou & Plionis method to the two groups. The top and bottom panels
are for galaxies in the inner region and outer region, respectively. The left
panels are the results of the Hwang & Lee method, with black dots indicating
the velocities of individual galaxies and the solid line indicating the fitted
sinusoidal curve (Equation 3). The right panels show the results of the
Manolopoulou & Plionis method. Although the top left and bottom right panels
seem to show a sign of rotation, other results do not. The two methods give
different results because of their intrinsic characteristic. For example, the
Hwang & Lee method assumes that the line-of-sight velocity of a rotating
galaxy cluster follows a sinusoidal relation. On the other hand, the
Manolooulou & Plionis method may suffer from a binning effect on the position
angle, especially when the cluster has a weak rotation signal.

To make a quantitative assessment, we plot the rotational velocity and
position angle as a function of distance in Figure 11. The top panel shows the
rotation amplitude $v_{rot}$ and the bottom panel shows the position angle of
the rotation axis, measured from north to east. The blue triangles indicate
the results following the Hwang & Lee method, while the red circles represent
the results using the Manolopoulou & Plionis method. The error bars are
$1\sigma$ resampling uncertainties. The ratios between the rotational velocity
amplitude and the corresponding uncertainty are 3.12 (inner region) and 1.53
(outer region) for the Hwang & Lee method and 0.47 (inner region) and 1.86
(outer region) for the Manolopoulou & Plionis method. Although the signal-to-
noise ratio for the Hwang & Lee method in the inner region is relatively high,
other results do not present a definite signal of global rotation.

In addition, we apply a nonparametric test for detecting the rotation. We use
a circular-linear correlation test (Mardia & Jupp, 2000), which tests if there
is a correlation between a circular variable and a linear variable. A
directional variable represents the direction and is in units of degrees or
radians. That is, 0° and 360° are treated as equal values. A linear variable
does not have this property; examples include velocity and size. We utilize
the R package Directional, which calculates the $p$ value of the null
hypothesis of zero correlation between a directional variable and a linear
variable. We apply the hypothesis test to the subsamples mentioned above. The
$p$ values of zero correlation are 0.908 and 0.0001 for the inner region and
the outer region, respectively. For the inner region, the null hypothesis
cannot be rejected even at a 68% confidence level. The null hypothesis for the
outer region can be rejected at a 99% confidence level; however, the existence
of a correlation does not necessarily lead to the existence of rotation. One
counterexample is the case where there is a bulk motion in one part with all
other regions showing zero velocity. Combining the results from the parametric
test and the nonparametric test, it is hard to say for sure that the Perseus
cluster is rotating within 60′.

####  3.2.3 Examining the Existence of Substructures

We check for signs of substructure in the Perseus cluster using the
$\delta$-test (Dressler & Shectman, 1988). The $\delta$-test uses radial
velocities and positional information of galaxies to quantify the local
deviation of mean velocity and dispersion from the global values. To be more
specific, the $\delta$ value of each galaxy is calculated as

$\delta^{2}=(N_{nn}/\sigma^{2})[(\bar{v}_{local}-\bar{v})^{2}+(\sigma_{local}-\sigma)^{2}]$
(4)

where $\bar{v}$ and $\sigma$ denote the mean and the standard deviation of the
radial velocities, respectively. Variables with the subscript $local$ denote
local values and those without subscripts indicate global values. The local
values are calculated with $N_{nn}$ nearest-neighbors. We use $N_{nn}=11$ as
suggested by Dressler & Shectman (1988). Figure 12 shows the results of
conducting the $\delta$-test with galaxies inside 30′ from cluster center and
brighter than 19.1 in $r$-band, which are the conditions where the 2D
completeness is high and uniform. Each circle corresponds to a galaxy, with
the radius of the circle proportional to $e^{\delta}$. The outer dotted circle
shows the 30′ boundary. A clump of large circles in the figure would indicate
a possible substructure. A sparse clump exists at the northwest part of the
cluster, which was also noted by Meusinger et al. (2020).

Figure 12:  Distribution of galaxies brighter than 19.1 mag within 30′. The
radius of each circle is proportional to $e^{\delta}$. The cluster center is
marked with the red cross.

To examine whether such substructures exist, we conduct a Monte-Carlo
simulation of the $\Delta$-test (Dressler & Shectman, 1988), where the
$\Delta$ statistic is the sum of all $\delta$. We first calculate $\Delta$ of
the current data, $\Delta_{obs}$. Then, we create 5000 simulated clusters by
randomly shuffling the radial velocities of galaxies with the positions fixed
and calculate $\Delta$ for each simulated cluster, $\Delta_{sim}$. The
simulated data will thus have the same positional property and velocity
histogram as the observed cluster but with a different combination of the two.
Figure 13 shows the histogram of $\Delta_{sim}$ with the value of
$\Delta_{obs}$ marked with the red vertical line. The fraction of
$\Delta_{sim}$ greater than $\Delta_{obs}$ indicates the significance of the
substructure in the observed cluster. For our data,
$f(\Delta_{sim}>\Delta_{obs})=20\%$, implying that it is statistically
insignificant to assert that the current position and velocity distribution
are induced from a substructure.

Figure 13:  Histogram of $\Delta$ from Monte-Carlo simulated clusters. The red
vertical line indicates the $\Delta$ value of the real cluster.

###  3.3 Comparison of Spatial Distribution and Kinematics between Gas and
Galaxies

As galaxies and ICM have different timescales for relaxation after a merger
(e.g., Roettiger & Flores, 2000), comparing the kinematics of the two
components can reveal some information about the merger history of the
cluster. To this end, we compare the velocity maps of galaxies and ICM. For
the comparison of kinematics, we use all member galaxies regardless of their
apparent magnitude. We use two data sets for the ICM velocities. The first one
is the X-ray observation results from the Hitomi mission. Hitomi Collaboration
et al. (2018) presented precise ICM velocity measurements in seven central
regions of the Perseus cluster. Their observation region covers only
$R\lesssim 5\arcmin$, so we use another ICM velocity measurement with a wider
field from XMM-Newton observations. Using XMM-Newton archive data, Sanders et
al. (2020) measured the velocities of ICM in 21 non-overlapping regions around
the center of the Perseus cluster. The radius of the total region is $\sim
30\arcmin$, which includes more galaxies compared to the velocity map of
Hitomi. Figure 14 shows the velocity maps of galaxies (left) and ICM (right).
We mark the region borders with black solid lines. For the left panel, we only
show the borders within which three or more galaxies exist. The Hitomi regions
lie in the central part, while the XMM-Newton regions are located at the outer
part. The empty region in the XMM-Newton results is due to the Cu hole of the
XMM-Newton detector, which is the region of the detector where energy scale
calibration could not be done (refer to Sanders et al., 2020 for details). We
label regions from Hitomi (XMM-Newton) as ‘H’ (‘X’) followed by the region
number. For example, we denote H6 for Region 6 in the Hitomi field and X21 for
Region 21 in the XMM-Newton field. We note that H6 and X2 are overlapping with
each other, with X2 including only one more member galaxy. We compute the
galaxy velocity map using member galaxies and a Gaussian weight with a
standard deviation of $1.5\arcmin$, which is the narrowest width of the XMM-
Newton regions.

Figure 14:  Galaxy velocity map (left) and ICM bulk velocity map from Hitomi
and XMM-Newton results (right). The galaxy velocity map is weighted with a
Gaussian kernel of standard deviation 1.5′. Circles represent individual
galaxies color-coded according to their velocities, and the black cross is the
center of the cluster.

For quantitative comparison including error bars, we plot the velocities of
the two components as a function of projected distance in Figure 15. The
bottom panel shows the velocities of galaxies (blue open circles), ICM from
Hitomi (green filled diamonds), and ICM from XMM-Newton (red filled squares).
Only regions with three or more galaxies are plotted. The error bars of Hitomi
data are from Table 4 of Hitomi Collaboration et al. (2018) and those of XMM-
Newton are statistical errors from Table 2 of Sanders et al. (2020). The
velocity errors of galaxies are calculated as $\sigma/\sqrt{N}$, where
$\sigma$ is the standard deviation of the velocities of galaxies in the region
and $N$ is the number of galaxies in the region. The top panel of Figure 15
shows the velocity difference in each region. The error bars are
$\sqrt{\sigma_{gal}^{2}+\sigma_{ICM}^{2}}$ where $\sigma_{gal}$ and
$\sigma_{ICM}$ are errors of galaxy velocity and ICM velocity, respectively.
There are three regions with a velocity difference greater than $3\sigma$: H3
($11.4\sigma$), H6 ($9.5\sigma$), and X2 ($7.8\sigma$). As H6 overlaps with
X2, there are effectively two regions with significant velocity differences,
each located at $R=2.3\arcmin$ and $R=4.7\arcmin$. In the bottom plot of
Figure 15, we plot the radial velocities of individual member galaxies
belonging to H3 and X2 (black open circles). This confirms that the velocity
difference between galaxies and the ICM is not dominated by outlier galaxies
in both regions. It is also notable that the ICM velocities of the two
overlapping regions, H6 and X2, agree well despite the use of different
instruments. This indicates that the velocity differences between galaxies and
ICM in the H3 and X2 are intrinsic features of the cluster, instead of
systematic offsets due to instrumental characteristics. Other than H3 and X2,
the kinematics of galaxies agree with that of the ICM within $3\sigma$.

Figure 15:  Comparison of galaxy velocities and ICM velocities for regions
with more than three galaxies. (Top) Radial velocity difference of the two
components. (Bottom) Radial velocities of galaxies (blue open circle), ICM
from Hitomi results (green filled diamond), and ICM from XMM-Newton results.
The black open circles in H3 and X2 indicate velocities of individual galaxies
in the region, demonstrating that a single outlier galaxy in the region does
not dominate the difference in the galaxy and ICM velocities.

Lastly, we examine the spatial correlation between galaxy distribution and
X-ray emission in $R<30\arcmin$ region. Specifically, we compare the galaxy
number density with the X-ray intensity map. Galaxy number density is
calculated using Gaussian kernel density estimation with the standard
deviation of each Gaussian kernel fixed to 10′. We use member galaxies with
$r_{\rm{Petro},0}\leq 19.1$. For the X-ray intensity map, we use the mosaic
image of XMM-Newton, retrieved from XMM-Newton
website222https://heasarc.gsfc.nasa.gov/docs/xmm/gallery/esas-
gallery/xmm_gal_science_perseus.html. Figure 16 shows the galaxy number
density contours (black line) overlaid on the X-ray intensity map. As detector
gaps exist in the X-ray image (e.g., empty blanks rightward of the cluster
center and bottom-leftward), we resort to qualitative comparison by visual
inspection. The overall morphology of the cluster traced by ICM and galaxies
looks similar; in the inner region ($R\lesssim 15\arcmin$), the X-ray
intensity and galaxy number density are elongated in the east-west direction.

Figure 16:  Contours of galaxy number density (black line) overlaid on the
X-ray intensity map. The galaxy number density is estimated with a Gaussian
kernel of standard deviation 10′.

##  4 Discussion and Conclusions

We compile and present a catalog of galaxies with measured redshifts in the
vicinity of the Perseus cluster, consisting of our new data from MMT/Hectospec
along with those from the literature. We have developed CausticSNUpy, a Python
implementation of the caustic technique for cluster membership determination,
and analyzed the kinematics of cluster galaxies.

We first note that 1) no confident signal of rotation in $R<60\arcmin$ is
detected, 2) the galaxies in the Perseus cluster do not show substructures
within 30′, and 3) the kinematics and distribution of galaxies and ICM agree
at large. All three features give hints about past mergers of the cluster. For
example, the global rotation of a cluster may be a result of a recent off-axis
merger (Hwang & Lee, 2007). Also, substructures within a galaxy cluster are
interpreted as infalling galaxy groups (Guennou et al., 2014). The existence
of a substructure is evidence of a recent merger and an unrelaxed state.
Finally, simulations show that galaxies and ICM have different timescales to
react to a merger (Roettiger & Flores, 2000; Shin et al., 2022). In this
sense, our results as a whole imply the lack of evidence of a recent merger
and suggest that the Perseus cluster is a relaxed cluster.

There were also previous studies that suggested the relaxation of the Perseus
cluster by examining the ICM. Mohr et al. (1993) noted the elongation of ICM
isophotes and interpreted it as an X-ray substructure and as a sign of an
unrelaxed cluster. However, Markevitch et al. (2001) reported that relaxed,
cool-core clusters can exhibit X-ray substructures. Indeed, Simionescu et al.
(2012), from X-ray image, and Sanders et al. (2020), on the basis of low
turbulence, saw the Perseus cluster as a relaxed system. In this regard, our
galaxy-based analysis is consistent with the results from ICM studies.

It is also interesting that galaxies in the two inner regions ($R<5\arcmin$)
have a significant ($>7\sigma$) difference in velocities from both the ICM and
the cluster center (Figure 15). This difference does not seem to be a global
phenomenon at that radius. The caustics in Figure 8 tell us that a velocity
around 1000km s-1 is typical at 5′. Tamura et al. (2014) noted this deviation
of galaxy velocity in X2, along with a similar trend in gas velocities. They
suggested that this velocity structure is a byproduct of a past minor merger.
Here, we point out that three among the four galaxies belonging to X2 are very
close together when projected on the sky, almost indistinguishable in Figure
14 (dark blue region west of the cluster center). The maximum separation
between the three galaxies is 20″, equivalent to a projected distance of
5$h^{-1}$ kpc. Thus, the three galaxies in X2 could be a physically bound
compact group. For H3, the four galaxies are relatively sparsely distributed,
with the minimum pair-wise separation being 54″ (13$h^{-1}$ kpc). The true
identities of the velocity offsets in both H3 and X2 need further
investigation.

We present a catalog of galaxies with redshift measurements in the central
region ($R<60\arcmin$) of the Perseus cluster. The catalog is a product of a
deep spectroscopic survey of 1283 galaxies by MMT/Hectospec. Among them, we
extract 1151 reliable redshifts using the Python code RVSNUpy. The catalog
consists of 1447 galaxies within 60′: 1148 from MMT/Hectospec, 265 from SDSS,
29 from NED, and 5 from Meusinger et al. (2020). The new data from
MMT/Hectospec greatly increases the spectroscopic completeness in the region.
Specifically, the r-band apparent magnitude limit where the differential
completeness drops below 50% is deepened from 16.1 to 18.0 for $R<60\arcmin$
and from 16.1 to 19.1 for $R<30\arcmin$. In addition, the spectroscopic
completeness is spatially uniform in $R<30\arcmin$ region for galaxies with
$r_{\rm{Petro},0}\leq 19.1$. We use this new data set to study the kinematics
of the Perseus cluster and summarize our primary results as follows.

  1. 1.

We have used CausticSNUpy (an open-source Python module for cluster membership
determination) and identified 418 member galaxies within $R<60\arcmin$. Among
them, there are 370 red members and 48 blue members.

  2. 2.

We compute the velocity dispersion profile of member galaxies. Future studies
could use this data to infer the velocity anisotropy or the mass profile of
the cluster. Obtaining a stellar velocity dispersion of NGC 1275 would be very
helpful for giving a tight constraint on the mass model of the cluster.

  3. 3.

We examine the global rotation of the cluster within 60′ from the cluster
center but find no promising signal.

  4. 4.

The Perseus cluster does not exhibit significant substructures confirmed by
the $\delta$ and $\Delta$ statistics.

  5. 5.

The overall kinematics and spatial distributions of galaxies and X-ray-
emitting gas agree well on large scales.

  6. 6.

On small scales, the kinematics of galaxies and ICM show significant
differences in the inner two regions of the cluster. The large velocity
difference between the two in region X2 ($\sim 1200\rm{km\ s^{-1}}$) could be
because of a compact galaxy group, while that in region H6 remains uncertain.
These need to be studied in more detail with more data.

These signatures imply that the Perseus cluster is a relaxed system and has
not experienced a recent merger in the relaxation timescale of galaxies.

Our extensive catalog of galaxies with spectroscopic redshifts will provide a
useful data set for studying the dynamics in the central region of the Perseus
cluster. The utility of our data will be bolstered when coupled with precise
X-ray spectroscopic observations in the future, such as XRISM (by Japan
Aerospace Exploration Agency; JAXA) and Athena (by European Space Agency;
ESA). We expect that these high-resolution X-ray missions will widen the
observed region by Hitomi, which was only operable for about a month (Hitomi
Collaboration et al., 2016).

## Acknowledgments

This work was supported by K-GMT Science Program (PID: UAO-G181-21B) funded
through Korean GMT Project operated by Korea Astronomy and Space Science
Institute (KASI). Observations reported here were obtained at the MMT
Observatory, a joint facility of the University of Arizona and the Smithsonian
Institution. HSH acknowledges the support by the National Research Foundation
of Korea (NRF) grant funded by the Korea government (MSIT) (No.
2021R1A2C1094577). HS acknowledges the support of the National Research
Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No.
2022R1A4A3031306). NH and BGP acknowledge the support by the Korea Astronomy
and Space Science Institute (KASI) grant funded by the Korean government
(MSIT; No. 2023-1-860-02, International Optical Observatory Project). Funding
for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan
Foundation, the U.S. Department of Energy Office of Science and the
Participating Institutions. SDSS-IV acknowledges support and resources from
the Center for High-Performance Computing at the University of Utah. The SDSS
web site is [www.sdss.org](http://www.sdss.org/).

SDSS-IV is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS Collaboration including the Brazilian
Participation Group, the Carnegie Institution for Science, Carnegie Mellon
University, the Chilean Participation Group, the French Participation Group,
Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de
Canarias, The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, Lawrence Berkeley
National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP),
MaxPlanck-Institut für Astronomie (MPIA Heidelberg), Max-PlanckInstitut für
Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik
(MPE), National Astronomical Observatories of China, New Mexico State
University, New York University, University of Notre Dame, Observatário
Nacional / MCTI, The Ohio State University, Pennsylvania State University,
Shanghai Astronomical Observatory, United Kingdom Participation Group,
Universidad Nacional Autónoma de México, University of Arizona, University of
Colorado Boulder, University of Oxford, University of Portsmouth, University
of Utah, University of Virginia, University of Washington, University of
Wisconsin, Vanderbilt University and Yale University.

##  Appendix A Caustic Technique

###  A.1 Process of the Caustic Technique

Here, we give an overview of the caustic technique described by Diaferio
(1999) and Serra et al. (2011). First, candidate members of the cluster are
selected using a hierarchical clustering algorithm. The algorithm iteratively
clusters galaxies into groups based on their 2D positions and velocities. More
specifically, the similarity between two galaxies $i,j$ are defined by their
binding energy

$E_{ij}=-G\frac{m_{i}m_{j}}{R_{p}}+\frac{1}{2}\frac{m_{i}m_{j}}{m_{i}+m_{j}}\Pi^{2}$
(A1)

where $m_{i}$ and $m_{j}$ are the mass of galaxy $i,j$ (fixed to
$10^{12}h^{-1}\mathrm{M_{\sun}}$), $R_{p}$ is the projected separation, and
$\Pi$ is the line-of-sight velocity difference. At first, each individual
galaxy is a group by itself. At each step, the hierarchical clustering
algorithm merges two groups with the highest similarity (i.e., the strongest
binding energy). The similarity between group $g_{\alpha}$ and $g_{\beta}$ are
defined as

$E_{\alpha\beta}=\min_{i\in g_{\alpha},j\in g_{\beta}}E_{i,j}$ (A2)

which is the minimum pairwise binding energy between $g_{\alpha}$ and
$g_{\beta}$. The hierarchical clustering iterates this grouping until all
galaxies are in a single group.

A dendrogram, also called a binary tree, showing the grouping history can be
constructed after the hierarchical clustering is done. The path from the root
to the leaves that follows the subtree with the most leaves hanging from each
junction is called the main branch. To shortlist the candidate members from
the binary tree, we need to cut the main branch at a certain node and choose
the galaxies hanging from that node. The criterion for selecting the threshold
node is the velocity dispersion of galaxies hanging from the node. During
hierarchical clustering, when a substructure or an interloper is grouped into
the main branch, the velocity dispersion would increase significantly.
Otherwise, the velocity dispersion maintains a constant value. Using this
property, the threshold node is chosen as the point on the main branch where
the velocity dispersion of galaxies changes abruptly. The detailed
implementation of detecting this change differs from Diaferio (1999) and Serra
et al. (2011). Note that these candidate members do not necessarily coincide
with the final members determined using the caustics.

If the cluster center is not given, it can be estimated using the candidate
members. The right ascension and declination of the cluster center are
determined as the peak of the number density of candidate members on the 2D
sky $(\alpha,\delta)$. The number density is calculated using kernel density
estimation. The median of the candidate members is used as the redshift of the
cluster center.

Next, we draw the redshift diagram using all galaxies. The Hubble parameter
$H_{0}$ is used to convert the projected distance to have the same physical
dimension as the velocity. Another parameter, $q$, is used to rescale the axes
and adjust the weights of each axis. Diaferio (1999) chooses to use $q=25$ to
match the typical uncertainties in the measured positions and measured
velocities. Using these two parameters, we can plot the candidate member
galaxies in the redshift space as $\mathbf{x}=(rH_{0}q,v)$ where $r$ is the
projected distance of a galaxy from the cluster center and $v$ is the radial
velocity relative to the cluster center.

We then estimate the number density of galaxies in the redshift space. The
galaxy number density $f_{q}(\mathbf{x})$ is estimated using adaptive kernel
density estimation, which gives different kernel sizes to galaxies. The number
density at point $\mathbf{x}$ is estimated as

$f_{q}(\mathbf{x})=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{h_{i}^{2}}K(\frac{\mathbf{x}-\mathbf{x}_{i}}{h_{i}})$
(A3)

where $N$ is the number of galaxies, $\mathbf{x}_{i}$ is the redshift space
coordinates of each galaxies, $K$ is the kernel, and $h_{i}$ is the adaptive
kernel width assigned to each galaxies. Diaferio (1999) chose to use the
triweight function as a kernel defined as

$K(\mathbf{x})=\begin{cases}\frac{4}{\pi}(1-|\mathbf{x}|^{2})^{3},&|\mathbf{x}|<1\\\
0,&\text{otherwise}\end{cases}$ (A4)

The determination of adaptive kernel width is described in detail by Diaferio
(1999).

After estimating the galaxy number density, we determine the caustic
amplitude. Ideally, the caustics would be the solution $f_{q}(r,v)=0$. In real
situations, however, there exist interlopers and escaping galaxies. Thus, a
threshold $\kappa$ is used so that the contour $f_{q}(r,v)=\kappa$ determines
the caustic location. In addition, the contour $f_{q}(r,v)=\kappa$ would not
generally be symmetric along $v=0$ in the redshift space. Because spherical
symmetry is assumed, we make the caustics symmetric. We decompose the solution
of $f_{q}(r,v)=\kappa$ into the upper part $v_{up}(r)$ ($v>0$) and the lower
part $v_{low}(r)$ ($v<0$). The upper caustics $\mathcal{V^{+}}(r)$, the lower
caustics $\mathcal{V^{-}}(r)$, and in turn the caustic amplitude
$\mathcal{A}(r)$ are then chosen as

$\mathcal{V}^{\pm}(r)=\pm\min\\{|v_{up}(r)|,|v_{low}(r)|\\}$ (A5)
$\mathcal{A}(r)=\frac{1}{2}[\mathcal{V}^{+}(r)-\mathcal{V}^{-}(r)].$ (A6)

In addition, simulated galaxy clusters show $d\ln\mathcal{A}/dr\lesssim 1/4$.
To reflect this fact, a point on the caustic lines with a logarithmic
derivative greater than $\zeta$ is replaced with a value that satisfies
$d\ln\mathcal{A}/dr=1/4$. Diaferio (1999) chose $\zeta=1$ while Serra et al.
(2011) used $\zeta=2$. The final cluster members are those within the
caustics.

One task remains, which is choosing the right threshold $\kappa$. The caustics
correspond to escape velocity at the given distance, thus $\kappa$ can be
chosen to minimize

$S(\kappa,R)=|\langle v^{2}_{esc}\rangle_{\kappa,R}-4\langle
v^{2}\rangle_{R}|^{2}$ (A7)

where $R$ is the mean projected distance of candidate members, $\langle
v^{2}_{esc}\rangle_{\kappa,R}=\int^{R}_{0}\mathcal{A}^{2}(r)\phi(r)dr/\int^{R}_{0}\phi(r)dr$
is the mean squared caustic amplitude within $R$, and $\phi(r)=\int
f_{q}(r,v)dv$. Although $S(\kappa)=0$ indicates that the cluster is in
dynamical equilibrium within the radius $R$, Serra et al. (2011) assert that
this optimization gives valid $\kappa$ regardless of the dynamical state of
the cluster.

The caustic technique can be summarized as the following steps:

  1. 1.

Find candidate cluster members using a hierarchical clustering algorithm.

  2. 2.

If necessary, find the cluster center by locating the peak of the galaxy
number density and calculating the mean of the candidate member redshifts.

  3. 3.

Draw the redshift diagram using all galaxies and rescale the two axes using
$H_{0}$ and $q$ so that they have the same units and are evenly weighted.

  4. 4.

Estimate the galaxy number density $f_{q}(\mathbf{x})$ using adaptive kernel
density estimation.

  5. 5.

Start with an arbitrary $\kappa$.

  6. 6.

For given $\kappa$, find the solution to $f_{q}(r,v)=\kappa$.

  7. 7.

Calculate the caustic amplitude from Equations (A5) and (A6).

  8. 8.

Evaluate Equation (A7).

  9. 9.

Change the value of $\kappa$ and repeat steps 6 through 8 until a minimum has
been found.

  10. 10.

Final member galaxies are selected as those within the caustics.

###  A.2 CausticSNUpy and Caustic App

Figure 17:  Comparison of the results by our CausticSNUpy and the traditional
Caustic App. (Top) Absolute difference in the caustic amplitude. (Bottom)
Redshift diagram showing the caustics determined by CausticSNUpy (red line)
and Caustic App (blue line). Purple crosses are galaxies identified as cluster
members by both CausticSNUpy and Caustic App. Red squares and blue triangles
are galaxies identified only by CausticSNUpy and Caustic App, respectively.
Black open circles are non-members as identified by both programs.

We test the consistency of the caustics determined by our code, CausticSNUpy,
to those from the existing software Caustic App. As our aim is to implement
the caustic technique described by Diaferio (1999) and Serra et al. (2011), we
hereinafter regard the result of Caustic App as the answer. We use two data
sets for comparison: the catalog presented by Song et al. (2017) for galaxy
cluster Abell 2107 (A2107) and the catalog in this paper for the Perseus
cluster (A426). The bottom panels of Figure 17 show the result of the caustic
technique by CausticSNUpy (red) and Caustic App (blue). The purple crosses
represent galaxies identified as members by both CausticSNUpy and Caustic App.
The red squares and the blue triangles are the galaxies determined as members
by CausticSNUpy but not by Caustic App and vice-versa. The top panels show the
difference in the caustic amplitudes calculated by the two programs.
Qualitatively, the caustics calculated by our CausticSNUpy have an overall
similar shape as those yielded by the existing Caustic App.

For quantitative assessment, we calculate the completeness and excess of
member determination compared to the Caustic App. For A2107, CausticSNUpy
identifies all member galaxies found by Caustic App but also includes 5 more
galaxies; the completeness is 100% with 1.8% excess identification. For the
Perseus cluster, the member galaxies identified by CausticSNUpy include 12
more galaxies while finding all member galaxies identified by Caustic App.
Thus, the completeness of CausticSNUpy is 100% while the excess identification
rate is 6.8% for the Perseus cluster. We also observe for both programs that
the caustic location changes noticeably when different settings are used, such
as the initial velocity cuts applied before starting the hierarchical
clustering. In addition, as Diaferio (1999) noted, the choice of $\kappa$ as
the threshold of number density can be adjusted manually. Taking these factors
into account, CausticSNUpy can act as a reliable open-source Python code to
run the caustic technique and as a substitute for Caustic App.

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# Rooted partitions and number-theoretic functions

Bruce E. Sagan
Department of Mathematics, Michigan State University,
East Lansing, MI 48824-1027, USA<EMAIL_ADDRESS>
(
Key Words: bijection, Euler toitent function, integer partition, Möbius
function,
AMS subject classification (2020): 11P84 (Primary) 11A25, 05A17, 05A19
(Secondary) )

###### Abstract

Recently, Merca and Schmidt proved a number of identities relating partitions
of an integer with two classic number-theoretic functions, namely the Möbius
function and Euler’s totient function. Their demonstrations were mainly
algebraic. We give bijective proofs of some of these results. Our main tools
are the concept of a rooted partition and an operation which we call the
direct sum of a partition and a rooted partition.

##  1 Introduction

We first define three of the most well-studied functions in number theory. Let
$n$ be a nonnegative integer and

$[n]=\\{1,2,\ldots,n\\}.$

A partition of $n$ is a weakly decreasing sequence
$\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{\ell})$ of positive integers
called parts satisfying $|\lambda|:=\sum_{i}\lambda_{i}=n$. In this case we
also write $\lambda\vdash n$. We also let

${\cal P}(n)=\\{\lambda\mid\lambda\vdash n\\}$

so that the partition function is

$p(n)=\\#{\cal P}(n).$

Furthermore, for $n\geq 1$, let

$\Phi(n)=\\{k\in[n]\mid\gcd(k,n)=1\\}$

from which we get Euler’s totient function

$\phi(n)=\\#\Phi(n)$

where we use the hash symbol for cardinality. Finally, still for $n\geq 1$,
the Möbius function is

$\mu(n)=\left\\{\begin{array}[]{ll}(-1)^{\delta(n)}&\mbox{if $n$ is square
free,}\\\ 0&\mbox{else,}\end{array}\right.$ (1)

where

$\delta(n)=\text{ number of distinct prime divisors of $n$}.$

In a pair of recent papers, Merca and Schmidt [MS18, MS19] proved formulas
relating $p(n)$, $\phi(n)$, and $\mu(n)$. In order to state them, we will need
some further notation. Let

$S_{k}(n)=\text{ number of $k$'s in all the partitions of $n$}.$

For example, if $n=4$ and $k=1$ then

${\cal P}(4)=\\{(4),\ (3,1),\ (2,2),\ (2,1,1),\ (1,1,1,1)\\}$

and, counting the number of ones in each partition,

$S_{1}(4)=0+1+0+2+4=7.$

These numbers had appeared earlier in the literature. In particular, they
feature in the following theorem which is due to Fine [Fin59] although it is
sometimes referred to as Stanley’s Theorem [Sta12, p. 117, Exercise 80]. For a
thorough history of this and related results see the article of Gilbert
[Gil15]. Note that the first sum in equation (2) is just $S_{k}(n)$.

######  Theorem 1.1 ([Gil15]).

Fix $k\geq 1$ and for a partition $\lambda$ let

$f_{k}(\lambda)=\text{ the number of parts equal to $k$ in $\lambda$},$

and

$g_{k}(\lambda)=\text{ the number of distinct parts of $\lambda$ which occur
at least $k$ times}.$

For any $n\geq 0$

$\sum_{\lambda\vdash n}f_{k}(\lambda)=\sum_{\lambda\vdash n}g_{k}(\lambda).$
(2)

We will have to consider bounding the size of the parts of a partition. So let

$S_{k}^{\geq r}(n)=\text{ number of $k$'s in all the partitions of $n$ with
smallest part at least $r$}.$

Note that $S_{k}^{\geq 1}(n)=S_{k}(n)$. Also note that Merca and Schmidt used
$S_{n,k}^{(r)}$ for what we are calling $S_{k}^{\geq r}(n)$. But we prefer our
notation as it helps to keep track of which variable is used for which
parameter. We can now state the four identities which we will prove
bijectively. The original demonstrations were mainly by manipulation of
generating functions.

######  Theorem 1.2 ([MS18, MS19]).

For $n\geq 0$ we have the following equalities.

  1. 1.

$\displaystyle S_{1}(n)=\sum_{k=2}^{n+1}\phi(k)S^{\geq 2}_{k}(n+1)$.

  2. 2.

$\displaystyle p(n)=\sum_{k=3}^{n+3}\frac{\phi(k)}{2}S_{k}^{\geq 3}(n+3)$.

  3. 3.

$\displaystyle p(n)=\sum_{k=1}^{n+1}S_{k}(n+1)\mu(k)$.

  4. 4.

$\displaystyle p(n)=-\sum_{k=2}^{n+2}\mu(k)S_{k}^{\geq 2}(n+2)$.

One of our crucial objects in constructing the necessary maps will be rooted
partitions. Call a partition $\lambda=(\lambda_{1},\ldots,\lambda_{s})$ of $n$
rooted if one of its parts, say one of the $k$’s, has been singled out. This
part is called the root and will be denoted $\hat{k}$. Note that if $\lambda$
has $m$ parts equal to $k$ then it has $m$ corresponding partitions rooted at
$k$ which are considered different and distinguished by the order of $\hat{k}$
among the $k$’s. For example, if $\lambda=(5,2,2,2,1,1)$ then there are three
ways to root $\lambda$ at $2$, namely

$(5,\hat{2},2,2,1,1),\ (5,2,\hat{2},2,1,1),\ \text{and }(5,2,2,\hat{2},1,1).$

Let

${\cal S}_{k}(n)=\\{\lambda\vdash n\mid\text{$\lambda$ is a rooted partition
rooted at $k$}\\}$

and, more generally

${\cal S}_{k}^{\geq r}(n)=\\{\lambda\in{\cal S}_{k}(n)\mid\text{$\lambda$ has
smallest part at least $r$}\\}.$

Clearly $\\#{\cal S}_{k}^{\geq r}(n)=S_{k}^{\geq r}(n)$ for all values of the
parameters.

An important tool for combining partitions are rooted partition is what we are
calling the direct sum. Let $\lambda,\nu$ be two partitions with at most one
of them rooted. Their direct sum $\lambda\oplus\mu$ is obtained by, for each
$k$, concatenating the string of $k$’s in $\lambda$ with the string of $k$’s
in $\mu$, including the $\hat{k}$ if one exists. For example

$(5,2,2,1)\oplus(4,4,2,\hat{2},2,1,1)=(5,4,4,2,2,2,\hat{2},2,1,1,1).$

Note that this operation is not commutative as

$(4,4,2,\hat{2},2,1,1)\oplus(5,2,2,1)=(5,4,4,2,\hat{2},2,2,2,1,1,1).$

The rest of this paper is organized as follows. In the next section we will
bijectively prove the first two equalities in Theorem 1.2 which relate
partitions and Euler’s function. Section 3 will be devoted to demonstrating
the last two identies between $p(n)$ and $\mu(n)$. Most proofs will be
followed by an example illustrating the maps involved.

##  2 Connections with Euler’s totient function

We begin with a bijective proof of Theorem 1.2 (a) which we restate here for
convenience.

######  Theorem 2.1 ([MS18]).

For $n\geq 0$ we have

$S_{1}(n)=\sum_{k=2}^{n+1}\phi(k)S^{\geq 2}_{k}(n+1).$

###### Proof.

It suffices to give a bijection $f$ from ${\cal S}_{1}(n)$ to

${\cal
S}^{\prime}(n+1)=\\{(\lambda^{\prime},r)\mid\text{$\lambda^{\prime}\in{\cal
S}_{k}^{\geq 2}(n+1)$ for some $k$ and $r\in\Phi(k)$}\\}.$

Given $\lambda\in{\cal P}_{1}(n)$, let $o$ be the number of ones in $\lambda$.
Also let $p$ be the position of $\hat{1}$, where we number the $1$ positions
as $1,2,\ldots,o$ from left to right. For ease of reading, we write

$g=\gcd(o+1,p).$

Define $(\lambda^{\prime},r)=f(\lambda)$ as follows. Write

$\lambda=\mu\oplus\omega.$

where $\omega$ contains all the $1$’s in $\lambda$, includng $\hat{1}$, and
$\mu$ is the rest of $\lambda$. We now define $\omega^{\prime}$ to be $g$
copies of the part $(o+1)/g$ with the first of these as a root, and let

$\lambda^{\prime}=\mu\oplus\omega^{\prime}.$

Finally, we let $r=p/g$.

We must check that $(\lambda^{\prime},r)\in{\cal S}^{\prime}(n+1)$. To show
that $\lambda^{\prime}\vdash n+1$, merely note that since $|\lambda|=n$ and
$|\omega|=o$ we have

$|\lambda^{\prime}|=|\mu|+|\omega^{\prime}|=|\mu|+g\cdot(o+1)/g=|\mu|+|\omega|+1=n+1.$

For the restriction on part sizes, note that $\mu$ only has parts of size at
least $2$ by definition. And since $p\leq o$ it must be that $g<o+1$. So the
parts in $\omega^{\prime}$ are all of size $(o+1)/g\geq 2$. Thus
$\lambda^{\prime}=\mu\oplus\omega^{\prime}$ has only parts greater than or
equal to $2$. Finally $r=p/g$ and the root of size $(o+1)/g$ are clearly
relatively prime by definition of $g$.

To show that this map is a bijection, we construct its inverse. Given
$(\lambda^{\prime},r)$ we write

$\lambda^{\prime}=\mu\oplus\omega^{\prime}$

where $\omega^{\prime}$ consists of $\hat{k}$ and all the $k$’s to its right
in $\lambda^{\prime}$. Let $s=|\omega^{\prime}|$ and construct
$\mu\oplus\omega$ where $\omega$ is $s-1$ copies of $1$. Finally, root the $1$
of this direct sum in position $rs/k$ to form $\lambda$. It is a simple matter
to check that this is a well-defined map and the inverse of $f$, so we leave
those details to the reader. ∎

To illustrate this construction, suppose that
$\lambda=(4,4,2,1,1,\hat{1},1,1)\in S_{1}(15)$. Then we have $o=5$ ones and
the position of $\hat{1}$ is $p=3$. It follows that $g=\gcd(o+1,p)=3$. Writing

$\lambda=(4,4,2)\oplus(1,1,\hat{1},1,1)$

we replace the second summand with $g=3$ copies of $(o+1)/g=2$ with the first
copy as the root to get

$\lambda^{\prime}=(4,4,2)\oplus(\hat{2},2,2)=(4,4,2,\hat{2},2,2).$ (3)

For the second component of the image, we let $r=p/g=1$.

To reverse the process, suppose we are given $(\lambda^{\prime},r)$ where
$\lambda^{\prime}$ is as in equation (3) and $r=1$. Then
$\omega^{\prime}=(\hat{2},2,2)$ so that $k=2$ and $s=|\omega^{\prime}|=6$.
Thus $\omega$ has $6-1=5$ ones and

$(4,4,2)\oplus(1,1,1,1,1)=(4,4,2,1,1,1,1,1).$

Finally we root the one in position $rs/k=3$ to obtain the original $\lambda$.

For the proof of Theorem 1.2 (b), let $n,k$ be positive integers and consider

$\Psi(n)=\\{k<n/2\mid\gcd(k,n)=1\\}.$

Note that $n$ and $k$ are relatively prime if and only if $n$ and $n-k$ are as
well. And for $n$ even, $n$ and $n/2$ are never coprime unless $n=2$. It
follows that for $n\geq 3$ we have $\\#\Psi(n)=\phi(n)/2$.

######  Theorem 2.2 ([MS18]).

For $n\geq 0$ we have

$p(n)=\sum_{k=3}^{n+3}\frac{\phi(k)}{2}S_{k}^{\geq 3}(n+3).$

###### Proof.

From the discussion just before the theorem, it suffices to give a bijection
$f$ between ${\cal P}(n)$ and

${\cal Q}^{\geq 3}(n+3)=\\{(\lambda^{\prime},r)\mid\text{
$\lambda^{\prime}\in{\cal S}_{k}^{\geq 3}(n+3)$ for some $k$, and
$r\in\Psi(k)$}\\}.$

Given $\lambda\in{\cal P}(n)$ we define $(\lambda^{\prime},r)=f(\lambda)$ as
follows. Write

$\lambda=\nu\oplus\sigma$

where $\sigma$ contains all the ones and twos of $\lambda$, and $\nu$ contains
the remaining parts. Note that if $\lambda$ has smallest part at least $3$
then we let $\sigma$ be the empty partition. Also let

$\displaystyle s$ $\displaystyle=|\sigma|+3,$ $\displaystyle t$
$\displaystyle=(\text{number of twos in $\sigma$})+1,$ $\displaystyle g$
$\displaystyle=\gcd(s,t).$

Now construct

$\sigma^{\prime}=(\hat{k},k,\ldots,k)$

where $k=s/g$ and the number of parts of $\sigma^{\prime}$ is $g$. Finally,
define

$\lambda^{\prime}=\nu\oplus\sigma^{\prime}$

and

$r=t/g.$

To verify that$f$ is well defined, we first show that
$\lambda^{\prime}\in{\cal Q}_{k}^{\geq 3}(n+3)$. The sum of the parts of
$\lambda^{\prime}$ is correct since

$|\lambda^{\prime}|=|\nu|+|\sigma^{\prime}|=|\nu|+(s/g)g=|\nu|+|\sigma|+3=n+3.$

We must also make sure that $\lambda^{\prime}=\nu\oplus\sigma^{\prime}$ has no
ones or twos. This is true of $\nu$ by definition, and for $\sigma^{\prime}$
it suffices to show that $k\geq 3$. The number of twos in $\sigma$ is at most
$|\sigma|/2$. Now easy arithmetic manipulations imply that $s/t>2$. It follows
that $k=s/g\geq 3$.

To finish the well-definedness demomstration, we must check that
$r\in\Psi(k)$. Since $r=t/g$ and $k=s/g$ it is cear that $r$ and $k$ are
relatively prime. We also need $r<k/2$. But from the previous paragraph
$s/t>2$. This is equivalent to $kg/(rg)>2$ which is the desired inequality.

We finish by constructing $f^{-1}$. Suppose we are given
$(\lambda^{\prime},r)\in{\cal Q}^{\geq 3}(n+3)$ with $\lambda^{\prime}$ rooted
at $k$. Write

$\lambda^{\prime}=\nu\oplus\sigma^{\prime}$

where $\sigma^{\prime}$ contains $\hat{k}$ and all the $k$’s after it.
Constuct a partition $\sigma$ of ones and twos where

$|\sigma|=|\sigma^{\prime}|-3$

and the number of twos in $\sigma$ is $r|\sigma^{\prime}|/k-1$. Finally, we
let

$\lambda=\nu\oplus\sigma.$

We leave the demonstration that $f^{-1}$ is well defined and the inverse of
$f$ to the reader. ∎

By way of illustration, suppose $\lambda=(4,4,3,2,2,1,1)$. Then we write

$\lambda=(4,4,3,2,2,1,1)=(4,4,3)\oplus(2,2,1,1).$

So $\sigma=(2,2,1,1)$ from which we get $s=|\sigma|+3=9$ and $t=3$, one more
than the number of twos in $\sigma$. We now construct $\sigma^{\prime}$ which
is going to have $g=\gcd(s,t)=3$ copies of $k=s/g=3$ with the first one as
root. Thus

$\lambda^{\prime}=(4,4,3)\oplus(\hat{3},3,3)=(4,4,3,\hat{3},3,3).$

Lastly, let $r=t/g=1$ to get the final pair

$(\lambda^{\prime},r)=((4,4,3,\hat{3},3,3),\ 1).$

We now apply the inverse map to the pair $(\lambda^{\prime},r)$ from the
previous paragraph. Write

$\lambda^{\prime}=(4,4,3,\hat{3},3,3)=(4,4,3)\oplus(\hat{3},3,3)$

so that $\sigma^{\prime}=(\hat{3},3,3)$ and $k=3$. We now construct $\sigma$
having only ones and twos with $|\sigma|=|\sigma^{\prime}|-3=6$. Since $r=1$,
the number of twos in $\sigma$ is to be $r|\sigma^{\prime}|/k-1=2$. This
forces $\sl=(2,2,1,1)$. Finally

$\lambda=(4,4,3)\oplus(2,2,1,1)=(4,4,3,2,2,1,1)$

bringing us back to where we started.

##  3 Connections with the Möbius function

In this section we will use sign-reversing involutions to construct our
bijections. We begin with a brief review of this technique. For more
information, see [Sag20, Section 2.2].

Let ${\cal S}$ be a finite set which is signed in that it comes with an
associated function

$\operatorname{sgn}:{\cal S}\rightarrow\\{1,-1\\}.$

An involution on ${\cal S}$ is a bijection $\iota:{\cal S}\rightarrow{\cal S}$
such that $\iota^{2}$ is the identity map on ${\cal S}$. It is well known that
an involution must decompose into cyclces $(s)$ of length one (that is, fixed
points), and $(s,t)$ of length two (that is, distinct elements $s,t$ with
$\iota(s)=t$ and $\iota(t)=s$). We let $\operatorname{Fix}\iota$ be the set of
fixed points of $\iota$. Call $\iota$ sign reversing if

  1. 1.

for each fixed point $(s)$ we have $\operatorname{sgn}s=1$, and

  2. 2.

for each $2$-cycle $(s,t)$ we have $\operatorname{sgn}s=-\operatorname{sgn}t$.

It is clear from the definitions that

$\sum_{s\in{\cal S}}\operatorname{sgn}s=\\#\operatorname{Fix}\iota$ (4)

since all the fixed points have positive sign and the signs of elements in
$2$-cycles cancel out.

As is usual when constructing sign-reversing involutions we will have two
cases, depending on some parameter, which perform inverse operations which
change signs. The parameter which we will use is

$\pi(n)=\left\\{\begin{array}[]{ll}\text{smallest prime dividing $n$}&\mbox{if
$n\geq 2$,}\\\ \infty&\mbox{if $n=1$,}\end{array}\right.$

where we consider $\infty>p$ for any prime $p$.

######  Theorem 3.1 ([MS19]).

For $n\geq 0$ we have

$p(n)=\sum_{k=1}^{n+1}S_{k}(n+1)\mu(k).$

###### Proof.

From equation (1), we can restrict the sum in the theorem to those $k$ which
are square free. Let

${\cal S}(n)=\\{\lambda\mid\text{$\lambda\vdash n$ is a partition rooted at a
square free part}\\}.$

We turn ${\cal S}(n)$ into a signed set by letting the sign of a partition
$\lambda$ with root $\hat{k}$ be

$\operatorname{sgn}(\lambda)=\mu(k)=(-1)^{\delta(k)}.$

Since the number of ways to root $\lambda$ at $k$ is the number of $k$’s in
$\lambda$ we have

$\\#{\cal S}(n+1)=\sum_{k}S_{k}(n+1)$

where the sum is over all square free $k$. Futhermore, there is a bijection
between the partitions $\mu$ of $n$ and the partitions of ${\cal S}(n+1)$
obtained by inserting a $\hat{1}$ at the end of $\mu$. Note that the signs of
the partitions in this subset are all $\mu(1)=1$ which is positive. So, by
equation (4), it suffices to produce a sign-reversing involution $\iota$ on
${\cal S}(n+1)$ with the rooted partitions just described as fixed points,

If $\lambda\in{\cal S}(n+1)$ with root $\hat{k}$ then let $m$ be the number of
parts equal to $k$ after and including $\hat{k}$. Write

$\lambda=\mu\oplus\kappa$

where $\kappa$ contains $\hat{k}$ and the copies of $k$ to its right. We now
have two cases for constructing $\lambda^{\prime}=\iota(\lambda)$ depending on
the the relative sizes of $\pi(k)$ and $\pi(m)$.

Consider any $\lambda\in{\cal S}(n+1)$ not ending with $\hat{1}$. It follows
that $\min\\{\pi(k),\pi(m)\\}\neq\infty$ which will make the following cases
well defined. For the first case, suppose that

$\pi(k)\leq\pi(m)$ (5)

and let

$k_{1}=k/\pi(k)\quad\mbox{and}\quad m_{1}=m\cdot\pi(k).$

Now construct $\kappa^{\prime}$ as the partition consisting of $m_{1}$ copies
of $k_{1}$, where the first is rooted, and let

$\lambda^{\prime}=\mu\oplus\kappa^{\prime}.$

For the second case, if we have

$\pi(k)>\pi(m)$

then we define

$k_{2}=k\cdot\pi(m)\quad\mbox{and}\quad m_{2}=m/\pi(m).$

This time $\lambda^{\prime}$ is constructed in exactly the same way as the
first case, just using $k_{2}$ and $m_{2}$ in place of $k_{1}$ and $m_{1}$,
resepctively.

We first need to prove that $\iota$ is well defined in that
$\lambda^{\prime}\in{\cal S}(n+1)$ and
$\operatorname{sgn}\lambda^{\prime}=-\operatorname{sgn}\lambda$. To show that
$\lambda^{\prime}\vdash n+1$ note that we are replacing parts summing to $km$
with parts summing to $k_{1}m_{1}$ or $k_{2}m_{2}$. But the definitions make
it clear that these three products are equal so that $\lambda^{\prime}$ and
$\lambda$ both partition $n+1$. We must also make sure that $k_{1}$ and
$k_{2}$ are square free. This is immediate for $k_{1}$ since it is a quotient
of $k$. And $k_{2}=k\cdot\pi(m)$ where $\pi(m)$ is a prime smaller than the
smallest prime dividing $k$. So $k_{2}$ is also square free. Finally,
multiplying or dividing by a prime changes the total number of prime factors
(with multiplicty) by one. So, since we only have numbers which are square
free, $\delta(k)$ will also change by exactly one and thus change the sign in
passing from $\lambda$ to $\lambda^{\prime}$.

There remains to show that $\iota^{2}(\lambda)=\lambda$. Let
$\lambda^{\prime}=\iota(\lambda)$. We have two cases resulting from the
definition of $\iota$. Since the proofs are similar, we will only do the one
where (5) holds. From this inequality and the fact that $k$ is square free we
see that

$\pi(k_{1})=\pi(k/\pi(k))>\pi(k)=\pi(m\cdot\pi(k))=\pi(m_{1}).$

So $\lambda^{\prime}$ is in the second case. It is now trivial to verify that
$\iota(\lambda^{\prime})$ yields $\lambda$. ∎

As an illustration of the previous involution, suppose
$\lambda=(3,3,2,\hat{2},2,2,1,1)$. Thus the root is at $k=2$ and there are
$m=3$ parts of that size after and including $\hat{2}$. Furthermore

$\lambda=(3,3,2,1,1)\oplus(\hat{2},2,2)$

which implies $\kappa=(\hat{2},2,2)$. Now $\pi(k)=2$ and $\pi(m)=3$ so we are
in the first case of the involution. Thus $k_{1}=k/\pi(k)=1$ and
$m_{1}=m\cdot\pi(k)=6$ with corresponding partition
$\kappa^{\prime}=(\hat{1},1,1,1,1,1)$. Finally

$\lambda^{\prime}=(3,3,2,1,1)\oplus(\hat{1},1,1,1,1,1)=(3,3,2,1,1,\hat{1},1,1,1,1,1).$

Consider what happens when we apply $\iota$ to the $\lambda^{\prime}$ just
given. Now the root is $k=1$ with $m=6$ ones after and including $\hat{1}$. We
factor

$\lambda^{\prime}=(3,3,2,1,1,\hat{1},1,1,1,1,1)=(3,3,2,1,1)\oplus(\hat{1},1,1,1,1,1)$

and let $\kappa^{\prime}=(\hat{1},1,1,1,1,1)$. In this situation
$\pi(k)=\infty$ and $\pi(m)=2$ so we are in the second case of the involution.
Letting $k_{2}=k\cdot\pi(m)=2$ and $m_{2}=m/\pi(m)=3$ yields the partition
$\kappa=(\hat{2},2,2)$. Finally,

$\lambda=(3,3,2,1,1)\oplus(\hat{2},2,2)=(3,3,2,\hat{2},2,2,1,1)$

returning us to the original partition.

The proof of the following result is much like that of the previous theorem,
so we only provide a sketch.

######  Theorem 3.2 ([MS19]).

For $n\geq 0$ we have

$p(n)=-\sum_{k=2}^{n+2}\mu(k)S_{k}^{\geq 2}(n+2).$

###### Proof.

We will prove the equality in the form

$p(n)+\sum_{k=2}^{n+2}\mu(k)S_{k}^{\geq 2}(n+2)=0$

So it suffices to give a sign-reversing involution $\iota$ without fixed
points on the set

${\cal Q}(n)={\cal P}(n)\uplus\\{\lambda\mid\text{$\lambda\in{\cal
S}_{k}^{\geq 2}(n+2)$ for some $k$ which is square free}\\}$

where

$\operatorname{sgn}\lambda=\left\\{\begin{array}[]{ll}1&\mbox{if
$\lambda\in{\cal P}(n)$,}\\\ (-1)^{\delta(k)}&\mbox{if $\lambda$ is rooted at
$k$.}\end{array}\right.$

The definition of $\iota$ is so similar to that of the map used in the
previous theorem that we will content ourselves with defining it and leave the
rest of the details to the reader. If $\lambda\in{\cal P}(n)$ then construct
$\lambda^{\prime}=\iota(\lambda)$ as follows. Write

$\lambda=\mu\oplus\omega$

where $\omega$ contains the ones in $\lambda$ or is empty if there are none.
Let

$m=|\omega|+2.$

Construct $\omega^{\prime}$ which consists of $m/\pi(m)$ copies of $\pi(m)$
with the first one its root. Finally define

$\lambda^{\prime}=\mu\oplus\omega^{\prime}.$

On the other hand, if $\lambda\in{\cal S}_{k}^{\geq 2}(n+2)$ then
$\lambda^{\prime}$ is constructed exactly as in the proof of Theorem 3.1. ∎

## Acknowledgement

We wish to than Mircea Merca for helpful discussions and references.

## References

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  * [MS18] Mircea Merca and Maxie D. Schmidt.  A partition identity related to Stanley’s theorem.  Amer. Math. Monthly, 125(10):929–933, 2018.
  * [MS19] Mircea Merca and Maxie D. Schmidt.  The partition function $p(n)$ in terms of the classical Möbius function.  Ramanujan J., 49(1):87–96, 2019.
  * [Sag20] Bruce E. Sagan.  Combinatorics: the art of counting, volume 210 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, [2020] ©2020.
  * [Sta12] Richard P. Stanley.  Enumerative combinatorics. Volume 1, volume 49 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, second edition, 2012.