5pt

# Increasing the Spatial Coverage of Atmospheric Aerosol Depth Measurements
Using Random Forest and Mean Filters

###### Abstract

Aerosols play a critical role in atmospheric chemistry, and affect clouds,
climate, and human health. However, the spatial coverage of satellite-derived
aerosol optical depth (AOD) products is limited by cloud cover, orbit
patterns, polar night, snow, and bright surfaces, which negatively impacts the
coverage and accuracy of particulate matter modeling and health studies
relying on air pollution characterization. We present a random forest model
trained to capture spatial dependence of AOD and produce higher coverage
through imputation. By combining the models with and without the mean filters,
we are able to create full-coverage high-resolution daily AOD in the
conterminous U.S., which can be used for aerosol estimation and other studies
leveraging air pollutant concentration levels.

Index Terms—  Aerosol Optical Depth, Random Forest, Imputation

00footnotetext: Copyright 2023 IEEE. Published in the 2023 IEEE International
Geoscience and Remote Sensing Symposium (IGARSS 2023), scheduled for 16 - 21
July, 2023 in Pasadena, California, USA. Personal use of this material is
permitted. However, permission to reprint/republish this material for
advertising or promotional purposes or for creating new collective works for
resale or redistribution to servers or lists, or to reuse any copyrighted
component of this work in other works, must be obtained from the IEEE.
Contact: Manager, Copyrights and Permissions / IEEE Service Center / 445 Hoes
Lane / P.O. Box 1331 / Piscataway, NJ 08855-1331, USA. Telephone: + Intl.
908-562-3966.

##  1 Introduction

Aerosols are suspensions of particles in gas, with sizes ranging from
$10^{-9}$ to $10^{-4}$m. They originate from various sources such as
urban/industrial emissions, biomass burning, gaseous precursors, sea salt, and
dust [1]. Aerosols have a crucial role in regional and global climate systems,
affecting radiation budget, cloud formation, atmospheric circulation, and
surface temperature [2]. Ground-level aerosols, particularly PM10 (Particulate
Matter with diameter $\leq 10\mu m$) and PM2.5 (diameter $\leq 2.5\mu m$), are
linked to adverse health effects like asthma, stroke, heart disease, and
pregnancy complications [3][4]. Hence, understanding aerosol variations is
vital for climate change and public health research. Aerosol Optical Depth
(AOD) measures aerosols in a column of air and is commonly used to estimate PM
levels [5][6]. While the AErosol RObotic NETwork (AERONET) provides
distributed observations of spectral AOD [7], the point-based observations
lack the high-resolution spatial coverage of remote sensing. Satellite-derived
AOD products provide a solution for high-resolution AOD retrieval, and the
multiangle implementation of atmospheric correction (MAIAC) AOD are commonly
used in characterizing spatiotemporal variations of aerosols [8] and
estimating surface PM levels [9]. However, the coverage of satellite-derived
AOD is limited by cloud cover, snow cover, and surface brightness [10].

A variety of methods have been developed to gap-fill AOD by incorporating
external information, including geostatistical methods and machine learning
(ML). ML methods show high performance and computing efficiency, but mainly
rely on the relationships in the feature space and ignore the spatial
dependence. In this study, we incorporate spatial information into a random
forest model to impute the missing AOD values and provide a high-resolution
full-coverage AOD. To increase the spatial coverage of AOD and utilize spatial
dependency, we apply mean filters to the AOD data. The available MAIAC AOD was
used as the target variable, and external information (including
meteorological variables, elevation, wildfire smoke, and NDVI) was used as
predictors. We assessed the AOD imputation performance between the random
forest model with and without mean filters.

##  2 Materials and Methods

###  2.1 Materials

####  2.1.1 Satellite-Retrieved Product

The MODIS MAIAC algorithm retrieves daily AOD at 1 km spatial resolution from
the MODIS Aqua and Terra sensor. Compared with old DT and DB algorithms, MAIAC
improves the coverage on both dark vegetation and bright deserts and increase
the spatial resolution. We obtained 16 MODIS Sinusoidal tiles of MAIAC AOD
covering the conterminous U.S. from 2005 to 2021 from the NASA
website111https://lpdaac.usgs.gov/products/mcd19a2v006/. The MCD19A2 AOD data
product contains two bands of AOD measurements: blue band AOD at $0.47\mu m$
and green band AOD at $0.55\mu m$. We used both as target variables in the
evaluation.

In addition, a MODIS Combined 16-Day Normalized Difference Vegetation Index
(NDVI) at 1 km resolution, which is generated from the MODIS/006/MCD43A4
surface reflectance composites, was retried via Google Earth Engine. To align
with the daily AOD image, the closest date’s NDVI image is associated with the
AOD.

####  2.1.2 Meteorological Data and Elevation Data

We obtained meteorological data from Daymet [11] and gridMET [12]. The Daymet
dataset includes minimum and maximum temperature, precipitation, shortwave
radiation, vapor pressure, day length. The dataset covers our study period
from 2005 to 2021, and is at a 1km $\times$ 1km spatial resolution and a daily
temporal resolution. To account for the wind impact on aerosol dispersion, we
also obtained wind direction and wind velocity variables from gridMET, a
dataset of daily high-spatial resolution ( 4 km, 1/24th degree) surface
meteorological data. The two gridMET variables are resampled and aligned with
other 1 km resolution variables.

A 1km resolution elevation dataset is obtained from EarthEnv
222http://www.earthenv.org/topography, which is derived from the digital
elevation model product of global 250m GMTED2010 and near-global 90 m
SRTM4.1dev [13].

####  2.1.3 Wildfire Smoke Data

NOAA Satellite Analysis Branch’s Hazard Mapping System (HMS) combines near
real-time polar and geostationary satellite observations into a common
framework in which expert image analysts perform quality control of automated
fire products and digitization of smoke plumes. HMS’s smoke analysis is based
on visual classification of plumes using GOES-16 and GOES-17 ABI true-color
imagery, and the smoke attributes are used to outline the smoke polygon and
plume density. The density information is qualitatively described using light,
medium, and heavy labels that are assigned based on the apparent thickness
(opacity) of the smoke in the satellite imagery.

The daily wildfire smoke polygon dataset is retrieved from NOAA
website333https://www.ospo.noaa.gov/Products/land/hms.html#data, and the data
available date is from August 5th, 2005, until now.

####  2.1.4 Spatial and Temporal Encoding

The spatial and temporal encoding variables included the coordinate (latitude
and longitude), year, and temporal cyclical features encoding (Cosine and Sine
of month and day of the year). The spatial and temporal encodings are designed
to capture the spatial and temporal variability of AOD.

###  2.2 Methods

####  2.2.1 Mean Filters

A mean filter is commonly used in image processing, and is designed to reduce
the amount of intensity variations between neighboring pixel values and reduce
noise in images. We use mean filters to increase the spatial coverage of AOD
data with high rates of missingness, and capture spatial information of
neighboring pixels based on the first law of geography [14].

Mean filtering can be considered a convolution filter, and the average of the
values in the local neighborhood replaces the value of each pixel. This is
presented as the following equation:

$g(x,y)=\frac{1}{n}\sum_{(i,j)\in S}f(i,j)$

where $g(x,y)$ is the smoothed image derived from the original image $f(i,j)$.
$S$ is the neighborhood set of $(x,y)$ and $n$ is the number of none-null
values in $S$. Fig.1 shows an example of a mean-filtered AOD image.

Fig. 1: Example of Mean Filters. (a) Raw green band AOD. (b) Mean-filtered
green band AOD

We applied 11$\times$11 mean filters to the MAIAC AOD data for both green and
blue bands, and gained about $15\%$ increase in the spatial coverage.

####  2.2.2 Random Forest

Random Forest (RF) is an ensemble learning algorithm that combines multiple
base learners to make collective predictions [15]. By aggregating the outputs
of several individual models, RF can capture non-linear relationships and
enhance the algorithm’s accuracy and generalization capabilities. Recently, it
has been commonly used in AOD imputation and PM2.5 concentration estimations
[16]. In this study, we employed the RF with mean filters to impute daily AOD
values for the conterminous U.S., showing the improvement of the mean filters.

The construction of RF regression model involves two primary steps: (1) random
subspace selection and (2) ensemble aggregation. For (1), a subset of features
is randomly selected for each decision tree and such randomness promotes
diversity among the trees, enabling the model to capture different aspects of
the data. In (2), the predictions from individual decision trees are combined
using averaging to produce the final prediction. The ensemble strategy
enhances the model’s stability, accuracy, and ability to handle high-
dimensional data.

Table 1:  Comparison of Model Performance for MAIAC AOD Imputation. | Random Validation | Spatial Validation | Temporal Validation
---|---|---|---
| R2 | RMSE | MBE | R2 | RMSE | MBE | R2 | RMSE | MBE
AOD 047 with mean filters | 0.97 | 36.23 | 0.10 | 0.96 | 40.91 | -0.40 | 0.94 | 36.44 | -0.31
AOD 047 w/o mean filters | 0.61 | 132.08 | -0.09 | 0.48 | 138.21 | 1.78 | 0.29 | 151.54 | -17.53
AOD 055 with mean filters | 0.97 | 25.97 | 0.07 | 0.95 | 28.20 | 1.08 | 0.95 | 33.33 | -0.82
AOD 055 w/o mean filters | 0.63 | 94.81 | 0.08 | 0.42 | 124.45 | -2.02 | 0.30 | 111.53 | -7.85

####  2.2.3 Training and Evaluation

For each day from August 5th, 2005, to December 31st, 2021, data points
located at EPA’s monitoring stations are used to train the RF models. The
MAIAC AOD at blue and green bands are set as target variables and multiplied
by a scale factor of 1000. We used random search cross-validation (CV) to
fine-tune the hyperparameters of RF, including the number of trees, the
maximum depth of the tree, the minimum number of samples required to split an
internal node, the minimum number of samples required to be at a leaf node,
and the number of features to consider when looking for the best split.

To better validate the performance of the model, three kinds of 5-fold CV were
deployed, including random CV, spatial CV, and temporal CV. In random CV,
samples are randomly divided into training and testing. 80% of samples were
used for model training and hyperparameters tuning, and the remaining samples
were held out for independent validation. Spatial CV separates the dataset
based on the location of the sites to account for spatial autocorrelation and
help prevent overfitting. Similarly, temporal CV divides the dataset into
training and testing according to the date, and accounts for temporal
autocorrelation.

In the evaluation, the coefficient of determination ($R^{2}$), Root Mean Squre
Error (RMSE), and Mean Bias Error (MBE) were used as the performance metrics.

##  3 Results and Discussions

During the study period, the daily coverage of MAIAC AOD in the conterminous
U.S. remained at an average of approximately 40%. The highest coverage month
was September, reaching 59%, and the lowest was February, with a coverage rate
of 29%. After applying the $11\times 11$ mean filters, the coverage increased
to 76% in September and 41% in Febuarary. The coverage of AOD has a high
spatial heterogeneity, with higher rates in states having fewer cloudy days
(e.g., California, Arizona, and New Mexico) and lower rates in northern states
(e.g., Maine, Michigan, and Minnesota).

Imputed AOD values of two bands from models with and without mean filters were
separately compared with MAIAC AOD retrievals to quantify the model
performance. Table 1 summarizes the statistics of held-out validation results.
Overall, the results of both blue and green bands demonstrate that the models
with mean filters are much better than models without mean filters in all the
evaluation metrics, including in spatial and temporal validation, with a much
higher $R^{2}$, lower RMSE, and MBE closer to MBE. These results indicate that
the models with mean filters are more robust with spatial and temporal
autocorrelation and are less biased than those without mean filters. The
average AOD of neighboring pixels has been proven to be a critical feature in
the imputation. The neighboring average AOD of two bands ranks the top two in
the feature importance (about 0.41), followed by the wildfire smoke density
(about 0.06).

Fig. 2: Example of results in the conterminous U.S. on Jan 1st, 2014. (a) Raw
green band AOD. (b) Imputed green band AOD.

By overlapping the two model outputs, with and without the mean filters (for
places where mean-filtering does is unavailable due to lack of data in the
11x11 window), we can obtain the full coverage of AOD in the conterminous U.S.
(Fig.2)

##  4 Conclusions

We proposed a random forest method with mean filters to capture the spatial
dependence of AOD to impute the missing data of AOD. According to the random,
spatial and temporal validation, the models with mean filters are confirmed to
have higher performance and less bias than the models without mean filters. By
combining the models with and without mean filters, we can obtain full spatial
coverage AOD in the conterminous U.S. The study has important implications for
the imputation of AOD and extensive applications of high-resolution MAIAC AOD
data.

##  5 Acknowledgments

This work was supported through NIH grant R21ES032973, and supported through
the University of Colorado Population Center (CUPC) funded by Eunice Kennedy
Shriver National Institute of Child Health & Human Development of the National
Institutes of Health (P2CHD066613).

## References

  * [1] G. Myhre, C. Myhre, B. Samset, and T. Storelvmo, “Aerosols and their relation to global climate and climate sensitivity,” _Nature Education Knowledge_ , vol. 4, no. 5, p. 7, 2013.
  * [2] Z. Li, X. Xia, M. Cribb, W. Mi, B. Holben, P. Wang, H. Chen, S.-C. Tsay, T. Eck, F. Zhao _et al._ , “Aerosol optical properties and their radiative effects in northern china,” _Journal of Geophysical Research: Atmospheres_ , vol. 112, no. D22, 2007.
  * [3] P. M. Mannucci and M. Franchini, “Health effects of ambient air pollution in developing countries,” _International journal of environmental research and public health_ , vol. 14, no. 9, p. 1048, 2017.
  * [4] X. Yu, M. M. Rahman, Z. Wang, S. A. Carter, J. Schwartz, Z. Chen, S. P. Eckel, D. Hackman, J.-C. Chen, A. H. Xiang _et al._ , “Evidence of susceptibility to autism risks associated with early life ambient air pollution: a systematic review,” _Environmental Research_ , vol. 208, p. 112590, 2022.
  * [5] J. Li, X. Ge, Q. He, and A. Abbas, “Aerosol optical depth (aod): spatial and temporal variations and association with meteorological covariates in taklimakan desert, china,” _PeerJ_ , vol. 9, p. e10542, 2021.
  * [6] M. Ghahremanloo, Y. Choi, A. Sayeed, A. K. Salman, S. Pan, and M. Amani, “Estimating daily high-resolution pm2. 5 concentrations over texas: Machine learning approach,” _Atmospheric Environment_ , vol. 247, p. 118209, 2021\.
  * [7] B. N. Holben, T. F. Eck, I. a. Slutsker, D. Tanré, J. Buis, A. Setzer, E. Vermote, J. A. Reagan, Y. Kaufman, T. Nakajima _et al._ , “Aeronet—a federated instrument network and data archive for aerosol characterization,” _Remote sensing of environment_ , vol. 66, no. 1, pp. 1–16, 1998.
  * [8] G. de Leeuw, C. Fan, Z. Li, J. Dong, Y. Li, Y. Ou, and S. Zhu, “Spatiotemporal variation and provincial scale differences of the aod across china during 2000–2021,” _Atmospheric Pollution Research_ , vol. 13, no. 4, p. 101359, 2022.
  * [9] Q. Di, I. Kloog, P. Koutrakis, A. Lyapustin, Y. Wang, and J. Schwartz, “Assessing pm2. 5 exposures with high spatiotemporal resolution across the continental united states,” _Environmental science & technology_, vol. 50, no. 9, pp. 4712–4721, 2016.
  * [10] A. Lyapustin and Y. Wang, “Modis multi-angle implementation of atmospheric correction (maiac) data user’s guide,” _NASA: Greenbelt, MD, USA_ , vol. 6, no. June, pp. 1–19, 2018.
  * [11] M. Thornton, R. Shrestha, Y. Wei, P. Thornton, S. Kao, and B. Wilson, “Daymet: daily surface weather data on a 1-km grid for north america, version 4 r1,” _ORNL DAAC, Oak Ridge, Tennessee, USA. Single Pixel Extraction Tool— Daymet (ornl. gov)_ , 2022.
  * [12] J. T. Abatzoglou, “Development of gridded surface meteorological data for ecological applications and modelling,” _International Journal of Climatology_ , vol. 33, no. 1, pp. 121–131, 2013.
  * [13] G. Amatulli, S. Domisch, M.-N. Tuanmu, B. Parmentier, A. Ranipeta, J. Malczyk, and W. Jetz, “A suite of global, cross-scale topographic variables for environmental and biodiversity modeling,” _Scientific data_ , vol. 5, no. 1, pp. 1–15, 2018.
  * [14] W. R. Tobler, “A computer movie simulating urban growth in the detroit region,” _Economic geography_ , vol. 46, no. sup1, pp. 234–240, 1970.
  * [15] L. Breiman, “Random forests,” _Machine learning_ , vol. 45, pp. 5–32, 2001\.
  * [16] Z.-Y. Chen, J.-Q. Jin, R. Zhang, T.-H. Zhang, J.-J. Chen, J. Yang, C.-Q. Ou, and Y. Guo, “Comparison of different missing-imputation methods for maiac (multiangle implementation of atmospheric correction) aod in estimating daily pm2. 5 levels,” _Remote Sensing_ , vol. 12, no. 18, p. 3008, 2020.


A generalization of immanants based on partition algebra characters

John M. Campbell

###### Abstract

We introduce a generalization of immanants of matrices, using partition
algebra characters in place of symmetric group characters. We prove that our
immanant-like function on square matrices, which we refer to as the
_recombinant_ , agrees with the usual definition for immanants for the special
case whereby the vacillating tableaux associated with the irreducible
characters correspond, according to the Bratteli diagram for partition algebra
representations, to the integer partition shapes for symmetric group
characters. In contrast to previously studied variants and generalizations of
immanants, as in Temperley–Lieb immanants and $f$-immanants, the sum that we
use to define recombinants is indexed by a full set of partition diagrams, as
opposed to permutations.

2020 Mathematics Subject Classification: 05E10, 15A15

immanant, partition algebra, character, irreducible representation

##  1 Introduction

The concept of the _immanant_ of a matrix was introduced in a seminal 1934
article by Littlewood and Richardson [19]. As suggested by Littlewood and
Richardson [19], by generalizing determinants and permanents of matrices using
symmetric group characters, this provides a way of unifying disparate areas of
combinatorial analysis, linear algebra, and representation theory. Since
partition algebras are such natural extensions of symmetric group algebras
[11], this leads us to consider how immanants of matrices may be generalized
using partition algebra characters. This forms the main purpose of our
article, in which we introduce the concept of the _recombinant_ of a matrix.
This gives us a generalization of immanants that is separate from the concept
of an $f$-immanant.

Given an $n\times n$ matrix

$A=\left(a_{i,j}\right)_{n\times
n}=\left(\begin{matrix}a_{1,1}&a_{1,2}&\cdots&a_{1,n}\\\
a_{2,1}&a_{2,2}&\cdots&a_{2,n}\\\ \vdots&\vdots&\ddots&\vdots\\\
a_{n,1}&a_{n,2}&\cdots&a_{n,n}\end{matrix}\right),$ (1)

the Leibniz identity for determinants is as below:

$\text{det}(A)=\sum_{\sigma\in
S_{n}}\left(\text{sgn}(\sigma)\prod_{i=1}^{n}a_{i,\sigma_{i}}\right),$ (2)

letting $S_{n}$ denotes the group of all permutations of $\\{1,2,\ldots,n\\}$.
The _permanent_ of (1) is defined by replacing the sign function in (2) as
below:

$\text{perm}(A)=\sum_{\sigma\in S_{n}}\prod_{i=1}^{n}a_{i,\sigma_{i}}.$ (3)

The matrix functions in (2) and (3) are special cases of the immanant function
defined in [19] and as below.

An integer partition is a finite tuple $\lambda$ of non-increasing natural
numbers. If the sum of all of the entries of $\lambda$ is a natural number
$n$, then $\lambda$ is said to be a partition of $n$, and this is denoted as
$\lambda\vdash n$. For $\lambda\vdash n$, we may let $\chi_{S_{n}}^{\lambda}$
be the irreducible character that is of the symmetric group $S_{n}$ and that
corresponds to $\lambda$. The _immanant_ $\text{Imm}^{\lambda}$ of (1) may be
defined so that:

$\text{Imm}^{\lambda}(A)=\sum_{\sigma\in
S_{n}}\chi^{\lambda}_{S_{n}}(\sigma)\prod_{i=1}^{n}a_{i,\sigma_{i}}.$ (4)

We find that the $\lambda=(1^{n})$ case of (4) agrees with (2) and the
$\lambda=(n)$ case of (4) agrees with (3). The purpose of this article is to
generalize (2), (3), and (4) using partition algebra characters, as opposed to
symmetric group characters.

Immanants are of interest within many different areas of advanced linear
algebra; see [2, 4, 5, 8, 12, 13, 15, 18, 26, 32], for example, and many
related references. The definition of immanants in terms of the irreducible
characters of the symmetric group naturally lends itself to applications
related to many different areas of algebraic combinatorics; for example, see
[1, 6, 7, 9, 17, 31] and many similar references. The foregoing considerations
reflect the interdisciplinary nature about immanants and motivate our
generalization of immanants.

Let $V$ denote an $r$-dimensional vector space. Let the general linear group
$\text{GL}_{r}(\mathbb{C})$ act on the tensor space $V^{\otimes n}$
diagonally. By taking $S_{r}$ as a subgroup of $\text{GL}_{r}(\mathbb{C})$ and
restricting the action of $\text{GL}_{r}(\mathbb{C})$ to permutation matrices,
partition algebras may be defined via the centralizer algebra

$P_{n}(r)\cong\text{End}_{S_{r}}\left(V^{\otimes n}\right),$ (5)

and the study of partition algebras had arisen within the field of statistical
mechanics via the centralizer algebra in (5), with reference to the work of
Jones [16] and Martin [21, 22, 23, 24]. This again speaks to the
interdisciplinary interest surrounding our generalization of immanants via
partition algebra characters.

###  1.1 Preliminaries

Our notation concerning partition algebras is mainly borrowed from Halverson’s
article on the character theory for partition algebras [10]. For the sake of
breivty, we assume familiarity with partition diagrams and the multiplication
of partition diagrams, referring to [10] for details. We let $P_{n}(r)$ denote
the $\mathbb{C}$-span of all order-$n$ partition diagrams, and we endow this
space with the multiplicative operation specified in [10]. Structures of this
form are referred to as _partition algebras_. We find that the symmetric group
algebra of order $n$ spanned by $\mathbb{C}$ is naturally a subalgebra, by
taking the span of partition diagrams of order $n$ with $n$ components with
exactly one vertex in the upper row and exactly one vertex in the lower row.

For integer partitions $\lambda$ and $\mu$, if $\mu_{i}\leq\lambda_{i}$ for
all $i$, then $\lambda/\mu$ denotes the skew shape obtained by removing $\mu$
from $\lambda$. We adopt the convention whereby the upper nodes of a partition
diagram of order $n$ are labeled with $1$, $2$, $\ldots$, $n$ and whereby the
lower nodes of this diagram are labeled with $1^{\prime}$, $2^{\prime}$,
$\ldots$, $n^{\prime}$. We then let $P_{n-1}(x)$ be embedded in $P_{n}(x)$ by
adding vertices labeled with $n$ and $n^{\prime}$ and by letting these
vertices be adjacent. From the branching rules subject to the restriction from
$P_{n}(r)$ to $P_{n-1}(r)$, and with the use of double centralizer theory via
(5), it can be shown that the irreducible representations of $P_{n}(r)$ are in
bijection with

$\widehat{P_{n}(r)}=\\{\lambda\vdash r\ :\ |\lambda^{\ast}|\leq n\\},$ (6)

where $\lambda^{\ast}=\lambda/(\lambda_{1})$.

We let $M^{\lambda}$ denote the irreducible representation of $P_{n}(r)$
indexed by $\lambda\in\widehat{P_{n}(r)}$. Following [10], we establish a
bijection between (6) and the set $\widehat{P_{n}}$ consisting of all
expressions of the form $\lambda^{\ast}$ in (6), i.e., by mapping $\lambda$ to
$\lambda^{\ast}$ and, conversely, by adding a row to $\lambda^{\ast}$
appropriately. For $\mu\in\widehat{P_{n}}$, we may let
$\chi_{P_{n}(x)}^{\lambda}$ denote the irreducible character of $P_{n}(x)$
corresponding to $M^{\lambda}$.

A basic result in the representation theory of groups is given by how
characters are constant on conjugacy classes. Halverson [10] introduced a
procedure for collecting partition diagrams so as to form analogues of
conjugacy classes, referring to [10] for details. For a diagram $d$, we let
$d_{\mu}$ denote the conjugacy class representative such that
$\chi(d)=\chi(d_{\mu})$ for a given partition algebra character $\chi$.

##  2 A generalization of immanants

For a permutation $p$ of order $n$ that we denote as a function

$p\colon\\{1,2,\ldots,n\\}\to\\{1,2,\ldots,n\\},$ (7)

we identify this permutation with the partition diagram corresponding to
$\\{\\{1,(p(1))^{\prime}\\}$, $\\{2,(p(2))^{\prime}\\}$, $\ldots$,
$\\{n,(p(n))^{\prime}\\}\\}$. We then consider this partition diagram as being
associated with the product

$\prod_{i=1}^{n}a_{i,p(i)},$ (8)

for the matrix $A$ in (1), and with regard to the summand in (4). So, this
raises the question as to what would be appropriate as an analogue of the
product in (8), for an _arbitrary_ partition diagram. This leads us toward the
following.

######  Definition 1.

For the $n\times n$ matrix in (1), we let the product $\prod_{d}a_{i,j}$ or
$\prod_{d}A$ be defined in the following manner. If $d$ is of propagation
number $0$, then we let the expression $\prod_{d}a_{i,j}$ vanish. If $d$ is of
a positive propagation number, let $B$ be a component of $d$ that is
propagating. We then form the product of all expressions of the form $a_{i,j}$
such that $i$ is in $B$ and $j^{\prime}$ is in $B$. Let $\Pi_{B}$ denote this
product we have defined using the component $B$. We then define
$\prod_{d}a_{i,j}$ as the product of all expressions of the form $\Pi_{B}$ for
all propagating components of $d$.

######  Example 1.

For the partition diagram

$d=\leavevmode\hbox to93.16pt{\vbox
to36.25pt{\pgfpicture\makeatletter\hbox{\hskip 3.9pt\lower-18.12637pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{88.85828pt}{-14.22638pt}\pgfsys@curveto{88.85828pt}{-12.29337pt}{87.29129pt}{-10.72638pt}{85.35828pt}{-10.72638pt}\pgfsys@curveto{83.42526pt}{-10.72638pt}{81.85828pt}{-12.29337pt}{81.85828pt}{-14.22638pt}\pgfsys@curveto{81.85828pt}{-16.1594pt}{83.42526pt}{-17.72638pt}{85.35828pt}{-17.72638pt}\pgfsys@curveto{87.29129pt}{-17.72638pt}{88.85828pt}{-16.1594pt}{88.85828pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{85.35828pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{85.35828pt}{-14.22638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{67.5187pt}{-14.22638pt}\pgfsys@curveto{67.5187pt}{-12.29337pt}{65.95172pt}{-10.72638pt}{64.0187pt}{-10.72638pt}\pgfsys@curveto{62.0857pt}{-10.72638pt}{60.5187pt}{-12.29337pt}{60.5187pt}{-14.22638pt}\pgfsys@curveto{60.5187pt}{-16.1594pt}{62.0857pt}{-17.72638pt}{64.0187pt}{-17.72638pt}\pgfsys@curveto{65.95172pt}{-17.72638pt}{67.5187pt}{-16.1594pt}{67.5187pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{64.0187pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{64.0187pt}{-14.22638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{46.17914pt}{14.22638pt}\pgfsys@curveto{46.17914pt}{16.1594pt}{44.61215pt}{17.72638pt}{42.67914pt}{17.72638pt}\pgfsys@curveto{40.74612pt}{17.72638pt}{39.17914pt}{16.1594pt}{39.17914pt}{14.22638pt}\pgfsys@curveto{39.17914pt}{12.29337pt}{40.74612pt}{10.72638pt}{42.67914pt}{10.72638pt}\pgfsys@curveto{44.61215pt}{10.72638pt}{46.17914pt}{12.29337pt}{46.17914pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{42.67914pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{42.67914pt}{14.22638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{88.85828pt}{14.22638pt}\pgfsys@curveto{88.85828pt}{16.1594pt}{87.29129pt}{17.72638pt}{85.35828pt}{17.72638pt}\pgfsys@curveto{83.42526pt}{17.72638pt}{81.85828pt}{16.1594pt}{81.85828pt}{14.22638pt}\pgfsys@curveto{81.85828pt}{12.29337pt}{83.42526pt}{10.72638pt}{85.35828pt}{10.72638pt}\pgfsys@curveto{87.29129pt}{10.72638pt}{88.85828pt}{12.29337pt}{88.85828pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{85.35828pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{85.35828pt}{14.22638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{46.17914pt}{-14.22638pt}\pgfsys@curveto{46.17914pt}{-12.29337pt}{44.61215pt}{-10.72638pt}{42.67914pt}{-10.72638pt}\pgfsys@curveto{40.74612pt}{-10.72638pt}{39.17914pt}{-12.29337pt}{39.17914pt}{-14.22638pt}\pgfsys@curveto{39.17914pt}{-16.1594pt}{40.74612pt}{-17.72638pt}{42.67914pt}{-17.72638pt}\pgfsys@curveto{44.61215pt}{-17.72638pt}{46.17914pt}{-16.1594pt}{46.17914pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{42.67914pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{42.67914pt}{-14.22638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{24.83957pt}{-14.22638pt}\pgfsys@curveto{24.83957pt}{-12.29337pt}{23.27258pt}{-10.72638pt}{21.33957pt}{-10.72638pt}\pgfsys@curveto{19.40656pt}{-10.72638pt}{17.83957pt}{-12.29337pt}{17.83957pt}{-14.22638pt}\pgfsys@curveto{17.83957pt}{-16.1594pt}{19.40656pt}{-17.72638pt}{21.33957pt}{-17.72638pt}\pgfsys@curveto{23.27258pt}{-17.72638pt}{24.83957pt}{-16.1594pt}{24.83957pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.33957pt}{-14.22638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.5pt}{-14.22638pt}\pgfsys@curveto{3.5pt}{-12.29337pt}{1.93301pt}{-10.72638pt}{0.0pt}{-10.72638pt}\pgfsys@curveto{-1.93301pt}{-10.72638pt}{-3.5pt}{-12.29337pt}{-3.5pt}{-14.22638pt}\pgfsys@curveto{-3.5pt}{-16.1594pt}{-1.93301pt}{-17.72638pt}{0.0pt}{-17.72638pt}\pgfsys@curveto{1.93301pt}{-17.72638pt}{3.5pt}{-16.1594pt}{3.5pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-14.22638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{24.83957pt}{14.22638pt}\pgfsys@curveto{24.83957pt}{16.1594pt}{23.27258pt}{17.72638pt}{21.33957pt}{17.72638pt}\pgfsys@curveto{19.40656pt}{17.72638pt}{17.83957pt}{16.1594pt}{17.83957pt}{14.22638pt}\pgfsys@curveto{17.83957pt}{12.29337pt}{19.40656pt}{10.72638pt}{21.33957pt}{10.72638pt}\pgfsys@curveto{23.27258pt}{10.72638pt}{24.83957pt}{12.29337pt}{24.83957pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.33957pt}{14.22638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.5pt}{14.22638pt}\pgfsys@curveto{3.5pt}{16.1594pt}{1.93301pt}{17.72638pt}{0.0pt}{17.72638pt}\pgfsys@curveto{-1.93301pt}{17.72638pt}{-3.5pt}{16.1594pt}{-3.5pt}{14.22638pt}\pgfsys@curveto{-3.5pt}{12.29337pt}{-1.93301pt}{10.72638pt}{0.0pt}{10.72638pt}\pgfsys@curveto{1.93301pt}{10.72638pt}{3.5pt}{12.29337pt}{3.5pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{14.22638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{67.5187pt}{14.22638pt}\pgfsys@curveto{67.5187pt}{16.1594pt}{65.95172pt}{17.72638pt}{64.0187pt}{17.72638pt}\pgfsys@curveto{62.0857pt}{17.72638pt}{60.5187pt}{16.1594pt}{60.5187pt}{14.22638pt}\pgfsys@curveto{60.5187pt}{12.29337pt}{62.0857pt}{10.72638pt}{64.0187pt}{10.72638pt}\pgfsys@curveto{65.95172pt}{10.72638pt}{67.5187pt}{12.29337pt}{67.5187pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{64.0187pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{64.0187pt}{14.22638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{{}{}}}{{}}{}{{{}{}}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{{{{{{}}{}{}{}{}{{}}}}}{}{}{}{}}{}\pgfsys@moveto{45.43684pt}{11.46867pt}\pgfsys@curveto{51.21504pt}{5.69046pt}{76.82236pt}{5.69046pt}{82.60057pt}{11.46867pt}\pgfsys@stroke\pgfsys@invoke{
}
{{}}{}{{{}{}}}{{}}{}{{{}{}}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{{{{{{}}{}{}{}{}{{}}}}}{}{}{}{}}{}\pgfsys@moveto{85.35828pt}{10.32639pt}\pgfsys@curveto{85.35828pt}{0.0pt}{85.35828pt}{0.0pt}{85.35828pt}{-10.32639pt}\pgfsys@stroke\pgfsys@invoke{
}
{{}}{}{{{}{}}}{{}}{}{{{}{}}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{{{{{{}}{}{}{}{}{{}}}}}{}{}{}{}}{}\pgfsys@moveto{82.60057pt}{-11.46867pt}\pgfsys@curveto{78.24509pt}{-7.11319pt}{71.1319pt}{-7.11319pt}{66.77641pt}{-11.46867pt}\pgfsys@stroke\pgfsys@invoke{
}
{{}}{}{{{}{}}}{{}}{}{{{}{}}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{{{{{{}}{}{}{}{}{{}}}}}{}{}{}{}}{}\pgfsys@moveto{61.6787pt}{-11.10638pt}\pgfsys@curveto{53.34892pt}{0.0pt}{53.34892pt}{0.0pt}{45.01915pt}{11.10638pt}\pgfsys@stroke\pgfsys@invoke{
}
{{}}{}{{{}{}}}{{}}{}{{{}{}}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{{{{{{}}{}{}{}{}{{}}}}}{}{}{}{}}{}\pgfsys@moveto{23.67958pt}{11.10638pt}\pgfsys@curveto{32.00935pt}{0.0pt}{32.00935pt}{0.0pt}{40.33913pt}{-11.10638pt}\pgfsys@stroke\pgfsys@invoke{
}
{{}}{}{{{}{}}}{{}}{}{{{}{}}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{{{{{{}}{}{}{}{}{{}}}}}{}{}{}{}}{}\pgfsys@moveto{39.92143pt}{-11.46867pt}\pgfsys@curveto{35.56595pt}{-7.11319pt}{28.45276pt}{-7.11319pt}{24.09727pt}{-11.46867pt}\pgfsys@stroke\pgfsys@invoke{
}
{{}}{}{{{}{}}}{{}}{}{{{}{}}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{{{{{{}}{}{}{}{}{{}}}}}{}{}{}{}}{}\pgfsys@moveto{18.58186pt}{-11.46867pt}\pgfsys@curveto{14.22638pt}{-7.11319pt}{7.11319pt}{-7.11319pt}{2.7577pt}{-11.46867pt}\pgfsys@stroke\pgfsys@invoke{
}
{{}}{}{{{}{}}}{{}}{}{{{}{}}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{{{{{{}}{}{}{}{}{{}}}}}{}{}{}{}}{}\pgfsys@moveto{2.34001pt}{-11.10638pt}\pgfsys@curveto{10.66978pt}{0.0pt}{10.66978pt}{0.0pt}{18.99956pt}{11.10638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

and for the $5\times 5$ case of (1), we find that

$\prod_{d}a_{i,j}=\prod_{d}A=\left(a_{2,1}a_{2,2}a_{2,3}\right)\left(a_{3,4}a_{3,5}a_{5,4}a_{5,5}\right).$

Definition 1 puts us in a position to offer a full definition for the concept
of the recombinant of a matrix, as below.

######  Definition 2.

We define the _recombinant_ of the square matrix in (1) so that

$\text{Rec}^{\lambda}(A)=\sum_{d\in
P_{n}(r)}\chi_{P_{n}(r)}^{\lambda}(d)\prod_{d}a_{i,j}.$ (9)

For example, an explicit evaluation for the recombinant, for non-propagating
submodules of partition algebras, of any $2\times 2$ matrix is given Section
2.1

Since our article is based on generalizing immanants using partition algebra
characters, it would be appropriate to prove, as below, that Definition 2 does
indeed generalize (4). In our below proof, we are to make use of the property
described by Halverson [10] whereby character tables for partition algebras
satisfy a recursion of the form

$\Xi_{P_{n}(x)}=\left[\begin{matrix}x\Xi_{P_{n-1}(x)}&\vdots&\ast\\\
\cdots&\hbox{}&\cdots\\\ 0&\vdots&\Xi_{S_{n}}\end{matrix}\right],$ (10)

where $\Xi_{S_{n}}$ denotes the character table of $S_{n}$.

By direct analogy with how Young tableaux are formed from paths in Young’s
lattice, _vacillating tableaux_ are formed from paths in the Bratteli diagram
$\hat{A}$ described in [11]. For the case whereby such a path ends on an
integer partition of order $n$ at level $n$ in $\hat{A}$, this corresponds to
an embedding of an irreducible representation of $\mathbb{C}S_{n}$ [11]. For a
vacillating tableau $T$ of this form, Theorem 1 below gives us that the
recombinant corresponding to the partition algebra representation $\rho$
corresponding to $T$ is the same as the immanant corresponding to the
symmetric group algebra representation corresponding to $\rho$.

######  Theorem 1.

For an $n\times n$ matrix $A$, if $|\lambda^{\ast}|=n$, then
$\text{\emph{Rec}}^{\lambda}(A)=\text{\emph{Imm}}^{\lambda^{\ast}}(A)$.

###### Proof.

First, let us write $\lambda\vdash r$ and $|\lambda^{\ast}|\leq n$, and let us
suppose that $\mu$ is a weak composition such that $0\leq|\mu|\leq n$. By
Corollary 4.2.3 from [10], we have that

$\chi_{P_{n}(r)}^{\lambda}\left(d_{\mu}\right)=0\ \ \ \text{if
$|\mu|<|\lambda^{\ast}|$,}$ (11)

and that the equality $|\mu|=|\lambda^{\ast}|=n$ implies that

$\chi_{P_{n}(r)}^{\lambda}\left(d_{\mu}\right)=\chi_{S_{n}}^{\lambda^{\ast}}\left(\gamma_{\mu}\right).$
(12)

For a permuting diagram $d$, Halverson’s procedure for conjugacy class
analogues [10] gives us that $\gamma_{\mu}$ is the cycle type for the
permutation corresponding to $d$, with $d=d_{\mu}$ written as a product of
disjoint, cyclic permutation diagrams. So, for an $n\times n$ matrix $A$ and
for $|\lambda^{\ast}|=n$, we find, from (11), that
$\chi_{P_{n}(r)}^{\lambda}(d)$ vanishes for all non-propagating partition
diagrams $d$, as in the lower left block of the character table in (10), so
that we may rewrite (9) so that

$\text{Rec}^{\lambda}(A)=\sum_{\text{prop}(d)=n}\chi_{P_{n}(r)}^{\lambda}(d)\prod_{d}a_{i,j},$
(13)

and where the character $\chi_{P_{n}(r)}^{\lambda}(d)$ reduces, in the manner
specified in (12), to the corresponding character of $S_{n}$ evaluated at the
permutation corresponding to the permuting diagram $d$. By Definition 1, the
product $\prod_{d}a_{i,j}$ in (13) is equal to $a_{1,d(1)}a_{2,d(2)}\cdots
a_{n,d(n)}$, writing the permuting diagram $d$ as a permutation as in (7). ∎

######  Remark 1.

Let us write $E_{\ell}$ to denote the partition diagram corresponding to

$\frac{1}{r}\\{\\{1,1^{\prime}\\},\\{2,2^{\prime}\\},\ldots,\\{\ell-1,(\ell-1)^{\prime}\\},\\{\ell,\ell+1,\ldots,n\\},\\{\ell^{\prime},(\ell+1)^{\prime},\ldots,n^{\prime}\\}\\}.$

We find that $P_{n}(r)E_{\ell}P_{n}(r)$ is a two-sided ideal and consists of
all linear combinations of partition diagrams with propagation number strictly
less than $\ell$. Fundamental results in the representation theory of
partition algebras are such that

$\mathbb{C}S_{n}\cong P_{n}(r)/\left(P_{n}(r)E_{n}P_{n}(r)\right)$ (14)

and such that any irreducible representation of $P_{n}(r)$ is either an
irreducible representation of $E_{n}P_{n}(r)E_{n}$ or an irreducible
representation of the right-hand side of (14); see [20, §4], for example, and
references therein. These properties can be used to formulate an alternative
proof of Theorem 1.

Our generalization of immanants, as above, is fundamentally different compared
to previously considered generalizations or variants of the immanant function.
Notably, Definition 2 is separate relative to how _$f$ -immanants_ are
defined. Following [28], an $f$-immanant, by analogy with (4), is of the form

$\text{Imm}^{f}(A)=\sum_{\sigma\in
S_{n}}f(\sigma)\prod_{i=1}^{n}a_{i,\sigma_{i}}$ (15)

for an arbitrary function $f\colon S_{n}\to\mathbb{C}$. A notable instance of
an $f$-immanant that is not of the form indicated in (4) is the
Kazhdan–Lusztig immanant, where the $f$-function in (15) is given by
Kazhdan–Lusztig polynomials associated to certain permutations. In contrast to
generalizations of immanants of the form shown in (15), our lifting of the
definition in (4) is based on a sum indexed by the diagram basis of
$P_{n}(r)$, in contrast to the index set for the sum in (15). In contrast to
immanants of $n\times n$ matrices being in correspondence with integer
partitions of $n$, and in contrast to $f$-immanants of $n\times n$ matrices
being in correspondence with class functions on $S_{n}$, we have that
recombinants of $n\times n$ matrices are in correspondence with the family of
integer partitions in (6).

###  2.1 An explicit evaluation

We find it convenient to denote partition algebra characters by writing
$\chi^{\lambda^{\ast}}(d)$ in place of $\chi_{P_{n}(r)}^{\lambda}(d)$.
Correspondingly, we may denote the recombinant associated with the character
$\chi^{\lambda^{\ast}}$ as $\text{Rec}^{\lambda^{\ast}}$. As below, we are to
let diagram basis elements be ordered according to the SageMath convention for
ordering such basis elements. According to this convention, let the diagram
basis of the order-$2$ partition diagram be ordered in the manner indicated in
Table 1, letting partition diagrams be denoted with set partitions.

$i$ | $d_{i}$ | $\chi^{\varnothing}(d_{i})$
---|---|---
1 |  $\\{\\{2^{\prime}$, $1^{\prime}$, $1$, $2\\}\\}$ | $1$
2 |  $\\{\\{2^{\prime}$, $1$, $2\\}$, $\\{1^{\prime}\\}\\}$ | $1$
3 |  $\\{\\{2^{\prime}\\}$, $\\{1^{\prime}$, $1$, $2\\}\\}$ | $1$
4 |  $\\{\\{2^{\prime}$, $1^{\prime}\\}$, $\\{1$, $2\\}\\}$ | $r$
5 |  $\\{\\{2^{\prime}\\}$, $\\{1^{\prime}\\}$, $\\{1$, $2\\}\\}$ | $r$
6 |  $\\{\\{2^{\prime}$, $1^{\prime}$, $1\\}$, $\\{2\\}\\}$ | $1$
7 |  $\\{\\{2^{\prime}$, $1\\}$, $\\{1^{\prime}$, $2\\}\\}$ | $2$
8 |  $\\{\\{2^{\prime}$, $1\\}$, $\\{1^{\prime}\\}$, $\\{2\\}\\}$ | $r$
9 |  $\\{\\{2^{\prime}$, $2\\}$, $\\{1^{\prime}$, $1\\}\\}$ | $2$
10 |  $\\{\\{2^{\prime}$, $1^{\prime}$, $2\\}$, $\\{1\\}\\}$ | $1$
11 |  $\\{\\{2^{\prime}$, $2\\}$, $\\{1^{\prime}\\}$, $\\{1\\}\\}$ | $r$
12 |  $\\{\\{2^{\prime}\\}$, $\\{1^{\prime}$, $1\\}$, $\\{2\\}\\}$ | $r$
13 |  $\\{\\{2^{\prime}\\}$, $\\{1^{\prime}$, $2\\}$, $\\{1\\}\\}$ | $r$
14 |  $\\{\\{2^{\prime}$, $1^{\prime}\\}$, $\\{1\\}$, $\\{2\\}\\}$ | $r$
15 |  $\\{\\{2^{\prime}\\}$, $\\{1^{\prime}\\}$, $\\{1\\}$, $\\{2\\}\\}$ | $r^{2}$
Table 1: The SageMath ordering for partition diagrams of order $2$, along with
the irreducible characters corresponding to non-propagating representations.

######  Example 2.

According to Definition 2, by writing

$\displaystyle\text{Rec}^{\varnothing}\left(\begin{matrix}a_{1,1}&a_{1,2}\\\
a_{2,1}&a_{2,2}\end{matrix}\right)=\sum_{d\in
P_{2}(r)}\chi^{\varnothing}(d)\prod_{d}a_{i,j}$
$\displaystyle=\chi^{\varnothing}(d_{1})\prod_{d_{1}}a_{i,j}+\chi^{\varnothing}(d_{2})\prod_{d_{2}}a_{i,j}+\cdots+\chi^{\varnothing}(d_{15})\prod_{d_{15}}a_{i,j},$

we may evaluate the recombinant $\text{Rec}^{\varnothing}$ according to the
character values shown in Table 1, so as to obtain that

$\displaystyle\text{Rec}^{\varnothing}\left(\begin{matrix}a_{1,1}&a_{1,2}\\\
a_{2,1}&a_{2,2}\end{matrix}\right)=$ $\displaystyle
a_{1,1}a_{1,2}a_{2,1}a_{2,2}+$ $\displaystyle
a_{1,1}a_{2,1}+a_{1,1}a_{1,2}+a_{1,2}a_{2,2}+a_{2,1}a_{2,2}+$ $\displaystyle
2\left(a_{1,1}a_{2,2}+a_{1,2}a_{2,1}\right)+$ $\displaystyle
r\left(a_{1,1}+a_{1,2}+a_{2,1}+a_{2,2}\right).$

For example, we may verify the above evaluation by computing the traces
associated with the linear transforms given by the action of left-
multiplication by diagram basis elements on the irreducible $P_{2}(r)$-module
$\mathscr{L}\\{d_{4},d_{14}\\}$.

We may obtain a similar evaluation, relative to Example 2, for the recombinant
that corresponds to the 3-dimensional representations of $P_{2}(r)$.

##  3 Conclusion

We conclude with some areas for future research concerning the matrix function
introduced in this paper.

A fundamental formula in algebraic combinatorics is Frobenius’ formula for
irreducible characters of the symmetric group, which, following [10], was
later shown by Schur to be a consequence of what is now know as _Schur–Weyl
duality_ between symmetric groups and general linear groups. The _irreducible
character basis_ introduced in [25] may be defined via a lifting of the
consequence

$p_{\mu}=\sum_{\lambda\vdash n}\chi_{S_{n}}^{\lambda}(\mu)s_{\lambda}$ (16)

of Schur–Weyl duality, with partition algebra characters used in place of
symmetric group characters in an analogue of (16). The SageMath implementation
of the $\tilde{s}$-basis from [25] provides a useful way of computing
partition algebra characters, which could be used to obtain a useful way of
computing recombinants. We encourage applications of this.

Temperley–Lieb algebras form an important family of subalgebras of partition
algebras. The _Temperley–Lieb immanants_ introduced by Rhoades and Skandera
[30] are $f$-immanants defined in a way related to Temperley–Lieb algebras,
referring to [30] for details. It seems that past research influenced by [30],
including relevant research on immanants or immanant-type functions as in [3,
27, 28, 29], has not involved any generalizations of immanants using partition
algebra characters. It may be worthwhile to explore relationships among
recombinants and Temperley–Lieb immanants, or to explore generalizations or
variants of recombinants related to the way Temperley–Lieb immanants are
defined.

The concept of a _twisted immanant_ was introduced in [14] and was based on
how the irreducible character $\chi^{\lambda}$, if restricted to an
alternating subgroup, splits as a sum of two irreducible characters, writing
$\chi^{\lambda}=\chi^{\lambda_{+}}+\chi^{\lambda_{-}}$. What would be an
appropriate notion of a _twisted recombinant_ , and how could this be applied
in a similar way, relative to [14]?

Immanants are often applied in the field of algebraic graph theory, via
immanants of Laplacian matrices and the like. How could recombinants be
applied similarly?

Immanants of Toeplitz matrices are often studied due to recursive properties
of such immanants. What is the recombinant of a given Toeplitz matrix?

### Acknowledgements

The author was supported through a Killam Postdoctoral Fellowship from the
Killam Trusts, and the author wants to thank Karl Dilcher for many useful
discussions. The author is thankful to Mike Zabrocki for useful comments
concerning the irreducible character basis and for many useful discussions
concerning partition algebras.

## References

  * [1] S. Clearman, B. Shelton, and M. Skandera, _Path tableaux and combinatorial interpretations of immanants for class functions on $S_{n}$_. In 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2011, pp. 233–244.
  * [2] M. P. Coelho and M. A. Duffner, _Subspaces where an immanant is convertible into its conjugate_. Linear and Multilinear Algebra 48(2001), no. 4, 383–408.
  * [3] H. de Guise, D. Spivak, J. Kulp, and I. Dhand, _$D$ -functions and immanants of unitary matrices and submatrices_. J. Phys. A 49(2016), no. 9, 09LT01, 12.
  * [4] M. A. Duffner, A. E. Guterman, and I. A. Spiridonov, _Converting immanants on skew-symmetric matrices_. Linear Algebra Appl. 618(2021), 76–96.
  * [5] C. Gamas, _Spherical functions and immanants_. Linear and Multilinear Algebra 47(2000), no. 2, 151–173.
  * [6] I. P. Goulden and D. M. Jackson, _Immanants of combinatorial matrices_. J. Algebra 148(1992), no. 2, 305–324.
  * [7] C. Greene, _Proof of a conjecture on immanants of the Jacobi-Trudi matrix_. Linear Algebra Appl. 171(1992), 65–79.
  * [8] R. Grone and R. Merris, _A Hadamard inequality for the second immanant_. J. Algebra 111(1987), no. 2, 343–346.
  * [9] M. Haiman, _Hecke algebra characters and immanant conjectures_. J. Amer. Math. Soc. 6(1993), no. 3, 569–595.
  * [10] T. Halverson, _Characters of the partition algebras_. J. Algebra 238(2001), no. 2, 502–533.
  * [11] T. Halverson and A. Ram, _Partition algebras_. European J. Combin. 26(2005), no. 6, 869–921.
  * [12] W. Hartmann, _On the complexity of immanants_. Linear and Multilinear Algebra 18(1985), no. 2, 127–140.
  * [13] P. Heyfron, _Some inequalities concerning immanants_. Math. Proc. Cambridge Philos. Soc. 109(1991), no. 1, 15–30.
  * [14] M. Itoh, _Twisted immanant and matrices with anticommuting entries_. Linear Multilinear Algebra 64(2016), no. 8, 1637–1653.
  * [15] G. James, _Hecke algebras and immanants_. Linear Algebra Appl. 197/198(1994), 659–670.
  * [16] V. F. R. Jones, _The Potts model and the symmetric group_. In Subfactors (Kyuzeso, 1993), World Sci. Publ., River Edge, NJ, 1994, pp. 259–267.
  * [17] M. Konvalinka, _On Goulden-Jackson’s determinantal formula for the immanant_. Ann. Comb. 13(2010), no. 4, 511–518.
  * [18] B. Kostant, _Immanant inequalities and $0$-weight spaces_. J. Amer. Math. Soc. 8(1995), no. 1, 181–186.
  * [19] D. E. Littlewood and A. R. Richardson, _Group characters and algebra_. Philos. Trans. R. Soc. Lond., Ser. A 233(1934), 99–142.
  * [20] C. Marcott, _Partition Algebras and Kronecker Coefficients_. M.Math. Thesis, University of Waterloo, 2015.
  * [21] P. P. Martin, _The partition algebra and the Potts model transfer matrix spectrum in high dimensions_. J. Phys. A 33(2000), no. 19, 3669–3695.
  * [22] P. Martin, _Potts models and related problems in statistical mechanics_. World Scientific Publishing Co., Inc., Teaneck, NJ, 1991.
  * [23] P. Martin, _Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction_. J. Knot Theory Ramifications 3(1994), no. 1, 51–82.
  * [24] P. Martin, _The structure of the partition algebras_. J. Algebra 183(1996), no. 2, 319–358.
  * [25] R. Orellana and M. Zabrocki, _Symmetric group characters as symmetric functions_. Adv. Math. 390(2021), Paper No. 107943, 34.
  * [26] T. H. Pate, _Tensor inequalities, $\xi$-functions and inequalities involving immanants_. Linear Algebra Appl. 295(1999), no. 1-3, 31–59.
  * [27] P. Pylyavskyy, _$A_{2}$ -web immanants_. Discrete Math. 310(2010), no. 15-16, 2183–2197.
  * [28] B. Rhoades and M. Skandera, _Kazhdan-Lusztig immanants and products of matrix minors_. J. Algebra 304(2006), no. 2, 793–811.
  * [29] B. Rhoades and M. Skandera, _Kazhdan-Lusztig immanants and products of matrix minors. II_. Linear Multilinear Algebra 58(2010), no. 1-2, 137–150.
  * [30] B. Rhoades and M. Skandera, _Temperley-Lieb immanants_. Ann. Comb. 9(2005), no. 4, 451–494.
  * [31] R. P. Stanley and J. R. Stembridge, _On immanants of Jacobi-Trudi matrices and permutations with restricted position_. J. Combin. Theory Ser. A 62(1993), no. 2, 261–279.
  * [32] R. Tabata, _Limiting behavior of immanants of certain correlation matrix_. Linear Algebra Appl. 510(2016), 230–245.

Department of Mathematics and Statistics

Dalhousie University

Halifax, NS, B3H 4R2

<EMAIL_ADDRESS>