# A poset version of Ramanujan results on Eulerian numbers and zeta values

Eric R Dolores-Cuenca1<EMAIL_ADDRESS>1Department of mathematics,
Yonsei University, Seoul, Korea  and  Jose L. Mendoza-Cortes2
<EMAIL_ADDRESS>2Department of Chemical Engineering & Materials Science,
Michigan State University, East Lansing, MI 48824, USA

###### Abstract.

We explore the operad of finite posets and its algebras. We use order
polytopes to investigate the combinatorial properties of zeta values. By
generalizing a family of zeta value identities, we demonstrate the
applicability of this approach. In addition, we offer new proofs of some of
Ramanujan’s results on the properties of Eulerian numbers, interpreting his
work as dealing with series inheriting the algebraic structure of disjoint
unions of points. Finally, we establish a connection between our findings and
the linear independence of zeta values.

##  1\. Introduction

During the final week of December 2021, the Mathematics Department at Yonsei
University organized a screening of math-related films. Among them was “The
Man Who Knew Infinity”, a biopic about the life of Ramanujan. Following the
screening, we headed to the library to peruse the copy [6] of “Ramanujan’s
Notebooks”.

Ramanujan’s notebooks do not feature his original proofs. As per [6], there
are various plausible reasons for this omission. Firstly, the notebooks were
intended to be a compilation of his findings rather than a comprehensive
guide. Secondly, given Ramanujan’s financial struggles, the exorbitant cost of
paper may have played a role. Additionally, it is feasible that Ramanujan
emulated the style of Carr’s book [17], which presents theorems with only
rough sketches of proofs. In an interview [36], S. Janaki, Ramanujan’s widow,
revealed that her husband “was fearful that English mathematicians would steal
his mathematical secrets while in England”.

Chapter 5 of Ramanujan’s first notebook left us astounded as we stumbled upon
the following declaration:

######  Entry 1.1 ([6, Chapter 5, Corollary 6] ).

Fix $n\in\mathbb{N}$. There are $\\{A_{i}\\}_{1\leq i\leq n}\in\mathbb{N}$
such that for all $1\leq r\leq n$:

$\displaystyle\sum_{k=1}^{r}A_{k}{r\choose k}$ $\displaystyle=$ $\displaystyle
r^{n}.$

The aforementioned theorem can be derived directly from the theory proposed in
our paper [3]. In the formula, $r^{n}$ represents the number of ways to label
the disjoint union of $n$ points using numbers between $1$ and $r$. The
coefficient $A_{k}$ counts a particular set of $k$–simplices in a given
triangulation of the $n$–cube. As we progressed through Chapter 5, we stumbled
upon several additional outcomes that align with our perspective.

####  1.0.1. What we do

We introduce the action of an operad on zeta values. An operad of sets
comprises a set of $n$-ary operations $O(n)$, with rules for composition and
actions of the symmetric group. These rules of composition represent an
abstraction of the composition of functions, while the action of the symmetric
groups reflects the permutations of inputs. Consequently, an algebra over an
operad $O$ is a set $X$ in which every element of $O(n)$ defines an operation
$X^{n}\rightarrow X$. For precise definitions, refer to [30, 34].

Consider an $n$-ary poset $P=\\{x_{1},\cdots,x_{n},<_{P}\\}$, which we
interpret as an operation, and let $P_{1},\ldots,P_{n}$ be posets. We define
the action of $P$ on $P_{1},\ldots,P_{n}$ to be the lexicographic sum of
$P_{1},\ldots,P_{n}$ over $P$. This means that we replace each vertex $x_{i}$
with the entire poset $P_{i}$, and then take the resulting poset with the
induced ordering. In other words, the operad of finite posets acts on the set
of finite posets.

The language of operads enables us to study algebraic properties that are
inherent to objects that have a defined composition, even if these objects do
not satisfy all the criteria of traditional algebraic structures like groups
or rings.

Our main result is the proof of the following identity. Let
$\zeta(k)=\sum_{n=1}^{\infty}\frac{1}{n^{k}}$ denote the Riemann zeta
function. Given $k\in\mathbb{N}$,

(1) $\displaystyle\sum_{n=k}^{\infty}(-1)^{n+1}{n\choose k}(\zeta(n+1)-1)$
$\displaystyle=$ $\displaystyle(-1)^{k+1}(\zeta(k+1)-1-1/2^{k+1}).$

This identity relates the alternating sum of a particular combination of
binomial coefficients and zeta values to a closed-form expression involving
the Riemann zeta function evaluated at integer arguments. Notably, when we
attempted to compute the left-hand side of Equation (1) using a variety of
public and private software packages in January 2023, we were unable to obtain
any output.

In the field of combinatorics, the order polynomial introduced by Stanley is a
polynomial that is associated with a finite poset. It is well known that
binomial polynomials serve as a basis for order polynomials. Our work shows
that order polynomials inherit the action of the operad of finite posets, and
that Equation (1) establishes a relationship between order polynomials and
zeta values as algebras over the operad of posets. This allows us to induce an
action of the operad of finite posets on zeta values.

We establish a connection between the theory of order polytopes and zeta
values, allowing us to compute results such as the following example: By
considering the cube as the union of 6 pyramids $\Delta[3]$, we can use the
inclusion-exclusion principle to account for the faces resulting from the
intersection of two pyramids. Thus, we have
$[0,1]^{3}=\sqcup_{6}\Delta[3]\setminus(\sqcup_{6}\Delta[2])\sqcup\Delta[1]$,
where the final term corresponds to the diagonal shared by all triangles
$\Delta[2]$ and all pyramids $\Delta[3]$. From this geometric interpretation,
we derive the identity

$\sum_{k=1}^{\infty}\frac{6{k\choose 3}-6{k\choose 2}+{k\choose
1}}{x^{k+1}}=\sum_{k=1}^{\infty}(-1)^{k+1}\frac{k^{3}}{x^{k+1}}=\frac{6}{(1-x)^{3+1}}-\frac{6}{(1-x)^{2+1}}+\frac{1}{(1-x)^{1+1}},$

which we express in finite form as

$\displaystyle\sum_{k=1}^{\infty}(-1)^{k+1}k^{3}(\zeta(k+1)-1)$
$\displaystyle=$
$\displaystyle{6}({\zeta(3+1)-1-\frac{1}{2^{3+1}}})-{6}({\zeta(2+1)-1-\frac{1}{2^{2+1}}})$
$\displaystyle+({\zeta(1+1)-1-\frac{1}{2^{1+1}}}).$

This illustrates the power of our approach in connecting seemingly disparate
areas of mathematics.

We demonstrate a novel technique for generating identities of zeta values from
specific triangulations of polytopes, establishing a connection between
enumerative combinatorics and number theory. Our approach is illustrated
through a detailed analysis of order polytopes, and we establish the validity
of this assignment for this class of polytopes. Notably, our work expands upon
the insights of Ramanujan. From our point of view, Ramanujan studied
identities linked to $n$-cubes.

Once we identified generators in an algebra over an operad, the action of the
operad can be described in the aforementioned basis via structural constants.
The action of the operad of posets on order polytopes is simple to describe,
we then compute the structural constants on polytopes and use those constants
to describe the action of the operad on zeta values. Our theorems yield
several results that are reminiscent of those discovered by Ramanujan.
Specifically, when we apply our theorems to the disjoint union of $n$ points
$P=\sqcup_{n}\langle 1\rangle$, we recover some of Ramanujan’s statements,
which were previously proven in the book [6]. Our proofs are distinct from
those given in [6] and, in some cases, shorter. To facilitate a comparison
with Ramanujan’s original work, we present his statements using the “Entry”
environment, as shown in Entry 1.1.

This paper is structured as follows: In Section 2, we delve into
combinatorics. We begin by presenting Lemma 2.3, which provides an
interpretation of the basis vectors $\\{{x\choose i}\\}_{i\in\mathbb{N}}$ in
terms of counting certain simplices in the canonical triangulation of the
order polytope of $P$, this result generalizes [3, Proposition 3.6]. Next, in
Theorem 2.4, we define the action of the operad of finite posets on Stanley
order polynomials. Additionally, we introduce the concept of order series, a
family of generating functions associated with a poset, and show that they
possess the structure of an algebra over the operad of finite posets,
extending the results of [3, Proposition 2.3]. We also explore the poset
$\\{x<y>z<w\\}$, which introduces a 4th-ary associative, non-commutative
operation. A series parallel poset is generated by the action of the
operations $\\{x<y\\}$, $\\{x,y\\}$ on chains $\langle
n\rangle=\\{1<2<\cdots<n\\}$. Our work highlights that the action of
$\\{x<y>z<w\\}$ is distinct from that of any poset generated by the operations
$\\{x<y\\}$, $\\{x,y\\}$, which expands upon the results of [3]. We provide
detailed explanations of our findings and include Lemma 2.11 to support our
claims.

In Section 3, we investigate posets that are formed by the disjoint union of
points. Our analysis leads to new proofs for certain results originally
presented by Ramanujan. Notably, our version of Corollary 3.3 is slightly more
general than the one found in [6].

In Section 4, we demonstrate the relationship between [6, Chapter 5, Example
1,2.] and [6, Chapter 5, Collolary 6 iv] with the disjoint union of points. To
extend these results to finite posets, we present Lemma 4.2 and Theorem 4.6.
Notably, Theorem 4.6 yields a shorter proof of [6, Chapter 5, Corollary 6.
iv)] when the poset is limited to be the disjoint union of points. Our paper’s
primary contribution is Theorem 4.6, which establishes Equation (1). Moreover,
the series of the form
$\sum_{i=1}^{\infty}a_{i}(\zeta(i+1)-1),a_{i}\in\mathbb{Q}$, are commonly
known as rational zeta series [7]. Our research expands the scope of rational
zeta series that we can compute.

In Section 5, we explore two algebraic structures of order polynomials/order
series: the vector space structure over $\mathbb{Z}$ and the algebraic
structure over the operad of posets. In Corollary 5.1, we demonstrate how the
vector space structure allows us to define the action of the operad of posets
on objects that are parameterized by binomial polynomials, such as zeta
values. Furthermore, we introduce a function $\tilde{n}$ that maps ${x\choose
k}$ to the value of Equation (1). We prove in Theorem 5.2 that the injectivity
of $\tilde{n}$ is equivalent to the linear independence of zeta values, which
is a prominent unsolved problem in number theory.

Section 6 delves into the potential connections between posets combinatorics
and some of the statements presented in Ramanujan’s notebooks. While no
explicit connection has been established previously, we discuss which
statements are likely to be explained with our approach.

##  2\. Order series

In this section, we venture into the realm of enumerative combinatorics,
building upon and generalizing the primary results of [3]. Throughout our
exploration, we will illustrate our approach with explicit poset examples.

For our analysis, we focus solely on finite posets and denote the poset
$1<2<\cdots<n$ as $\langle n\rangle$. Furthermore, we use $|P|$ to denote the
number of elements in a given poset $P$.

In his work [45], Stanley introduced the weak order polynomials $\Omega(P,n)$
and the strict order polynomials $\Omega^{\circ}(P,n)$. Specifically,
$\Omega(P,n)$ counts the number of non-strict order-preserving maps
$P\rightarrow\langle n\rangle$ such that $x<y$ implies $f(x)\leq f(y)$, while
$\Omega^{\circ}(P,n)$ counts the number of strict order-preserving maps
$P\rightarrow\langle n\rangle$ such that $x<y$ implies $f(x)<f(y)$.

In our analysis, we will utilize the following renowned result established by
Stanley:

######  Theorem 2.1 ([5, 49]).

Let $P$ be a poset. Then, we have

$(-1)^{|P|}\Omega^{\circ}(P,-x)=\Omega(P,x).$

This result is known as Stanley reciprocity.

######  Remark 2.2.

In order to establish a relationship between $\Omega^{\circ}(P,x)$ and
$\Omega(P,x)$ using Stanley reciprocity, we must first compute the coefficient
$(-1)^{|P|}$, which is dependent on the specific poset $P$. Therefore, simply
knowing the number of strict maps $hom_{\hbox{strict}}(P,n)$ is insufficient
for determining the number of weak maps $hom_{\hbox{weak}}(P,n)$; additional
information about the poset $P$ that induced the polynomials is required.

Below are some important properties of binomial coefficients that will be
utilized throughout the paper:

  * •

The binomial coefficient ${n\choose k}$ represents the number of ways to
choose $k$ items from a set of $n$ distinct objects.

  * •

If we expand the simplex $\Delta[k]$ $n$ times in each direction to form
$n\Delta[k]$, the number of lattice points in the simplex is ${n\choose k}$.

  * •

For a finite chain $\langle k\rangle$, we have $\Omega^{\circ}(\langle
k\rangle,n)={n\choose k}$, since the image of the order-preserving map can be
obtained by choosing $k$ different values from $n$.

  * •

The $n$-th coefficient of the generating function $\frac{x^{k}}{(1-x)^{k+1}}$
is ${n\choose k}$. This result can be derived from [23, Equation (1.3)] or
[51, Equation (1.5.5)].

Given a poset $P$, its order polytope $Poly(P)$ is a fundamental object in
order theory and combinatorial geometry [47]. Geometrically, we can associate
each element in $P$ with an axis in $[0,1]^{|P|}$ and define $Poly(P)$ as the
subspace of $[0,1]^{|P|}$ determined by the poset inequalities. Specifically,
$Poly(P)$ is the set of functions $f:P\rightarrow[0,1]$ that satisfy $f(x)\leq
f(y)$ whenever $x\leq_{P}y$. The canonical triangulation of $Poly(P)$ induced
by the poset structure decomposes $Poly(P)$ into $d_{|P|}$ simplices of
dimension $|P|$ that are glued together along $d_{|P|-1}$-simplices of
dimension $|P|-1$. However, we need to remove the extra copies of
$|P|-1$-simplices that were counted twice. For example, when $|P|=2$, we have
$Poly(P)=\Delta[2]\sqcup\Delta[2]\setminus\Delta[1]$. More generally, we can
write:

$Poly(P)=\sqcup_{d_{P,|P|}}\Delta[|P|]\setminus(\sqcup_{d_{P,|P|-1}}\Delta[|P|-1])\sqcup(\sqcup_{d_{P,|P|-2}}\Delta[|P|-2])\setminus\cdots(\sqcup_{d_{P,1}}\Delta[1]).$

We use this decomposition to define the coefficients $d_{P,i}$.

The strict order polynomials are integer-valued polynomials, and we can obtain
their Gregory-Newton decomposition [13, 14] as
$\Omega^{\circ}(P,x)=\sum_{i=0}^{|P|}a_{i}{x\choose i}$. The non-negativity
and integrality of the coefficients $a_{i}$ have been proven in [46] by
characterizing them as the number of surjective maps from $P$ into the chain
$\langle i\rangle$. In geometric terms, this quantity corresponds to the
number $d_{P,i}$ in the canonical triangulation of the order polytope
$Poly(P)$ [47]. This leads to an insightful interpretation of the binomial
basis vectors, which can be compared to the well-known $h^{*}$ vectors[48] and
$f^{*}$ vectors [9]:

######  Lemma 2.3 (Inclusion Exclusion).

Let $P$ be a finite poset, $n=|P|$. If the canonical triangulation of the
order polytope of $P$ is given by

$Poly(P)=\sqcup_{d_{P,|P|}}\Delta[|P|]\setminus(\sqcup_{d_{P,|P|-1}}\Delta[|P|-1])\sqcup(\sqcup_{d_{P,|P|-2}}\Delta[|P|-2])\setminus\cdots(\sqcup_{d_{P,1}}\Delta[1]),$

then, the strict order polynomial and the weak order polynomial of $P$ satisfy
$\Omega^{\circ}(P,x)=\sum_{i=j_{0}}^{n}d_{P,i}{x\choose i}$ and
$\Omega(P,x)=\sum_{i=1}^{n}(-1)^{|P|-i}d_{P,i}\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{x}{i}\right)\kern-3.00003pt\right)$.

The multiset
$\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{x}{n}\right)\kern-3.00003pt\right)={x+n-1\choose
n}$ is utilized in deriving the representation of $\Omega(P,x)$ from that of
$\Omega^{\circ}(P,x)$, through the application of Stanley reciprocity. One may
question the relevance of the multiset, which can be justified as follows:
consider a set of positive points denoted as $\\{a,b,\cdots,l|\\}$, and a set
of negative points denoted as $\\{|u,v,\cdots,z\\}$. The reason for the
appearance of the multiset in the enumeration of poset maps is that weak order
preserving maps $\langle k\rangle\rightarrow\langle n\rangle$ can be seen as
subsets of size $k$ from a set with $n$ negative elements
$\\{|1,2,\cdots,n\\}$. For further details, please refer to [18, 31], and [32]
which provide a new interpretation of Theorem 2.1. It is worth noting that
negative sets will not be used in the remainder of this paper.

In collaboration with Ayoub Youssefi Abqari (private communication on
10/31/2021), we have discovered that the results presented in Corollary 3,
Corollary 4, Theorem 3, and Theorem 4 of [24], which deal with iterated sums,
can all be derived through poset arguments. For instance, consider Corollary 4
of [24]:

######  Corollary ( 4 of [24]).

${n+k-q\choose
k}=\sum_{i_{k}=q}^{n}\sum_{i_{k-1}=q}^{i_{k}}\cdots\sum_{i_{2}=q}^{i_{3}}\sum_{i_{1}=q}^{i_{2}}1.$

###### Proof.

We may represent the potential values of a map from $\langle k\rangle$ to
$\langle n\rangle$ as $i_{1},\cdots,i_{k}$. The right-hand side of the
equation is counting the number of order-preserving maps from $\langle
k\rangle$ to $\langle n\rangle$ where every value is greater than or equal to
$q$. However, we can instead consider all non-strict maps from $\langle
k\rangle$ to $\langle n-(q-1)\rangle$, which amounts to $({n-q+1\choose
k})={n+k-q\choose k}$. ∎

After obtaining the vectors of $\Omega^{\circ}(P,x)$, we utilize Lemma 2.3 to
extract geometric information about the order polytope of $P$. As an
illustration, let us consider the poset $P=\\{x<y>z<w\\}$. The calculation of
$\Omega^{\circ}(P,x)$ yields ${x\choose 2}+5{x\choose 3}+5{x\choose 4}$. By
applying Lemma 2.3, we conclude that $Poly(P)$ can be constructed by
assembling 5 copies of the simplex $\Delta[4]$, where each simplex is attached
to two other $\Delta[4]$ along five $\Delta[3]$ simplices in such a way that
all $\Delta[4]$ simplex share a $\Delta[2]$ triangle.

Operads [30, 34] provide a powerful tool to handle customized operations.
Essentially, an operad of sets comprises a collection of $n-$ary operations
$O(n)$, governed by composition rules and acted upon by the symmetric group.
For instance, given a set $X$, there exists an operad $End_{X}$, where
$End_{X}(n)$ denotes the set of endomorphisms $X^{n}\rightarrow X$. In this
context, an algebra over an operad $O$ refers to an operadic morphism
$\varphi_{A}:O\rightarrow End_{A}$.

Consider a finite poset $P=\\{x_{1},\cdots,x_{|P|},\leq_{P}\\}$. To define
$P(P_{1},\cdots,P_{|P|})$, the Lexicographic sum [41, 50] is employed. This
operation involves taking the set $\sqcup P_{i}$ and imposing the order
$a\leq_{P(P_{1},\cdots,P_{|P|})}b$ if either $a\leq_{P_{i}}b$ for some $i$, or
if $a\in P_{r}$, $b\in P_{t}$, and $x_{r}\leq_{P}x_{t}$. For instance,
$\\{x<y\\}(\\{a,b\\},\\{c,d\\})=\\{a<c,a<d,b<c,b<d\\}$, as depicted in Figure
1. We denote the operad of finite posets by $\mathcal{FP}$.

${\\{c\\}}$${\\{d\\}}$${\\{a\\}}$${\\{b\\}}$ Figure 1. We present the
Lexicographic sum $\\{x<y\\}(\\{a,b\\},\\{c,d\\})$ of two posets $\\{a,b\\}$
and $\\{c,d\\}$, with the original poset $\\{x<y\\}$. An arrow from $r$ to $s$
implies that $r<s$. The resulting poset is constructed using the lexicographic
order, as defined in the main text.

######  Theorem 2.4.

The operad of finite posets acts on the Stanley order polynomials.

###### Proof.

To prove the theorem, we need to construct an operadic morphism
$\varphi_{A}:\mathcal{FP}\rightarrow End_{A}$, where $A$ is the set of strict
order polynomials indexed by posets. Consider a poset $P$ with $|P|$ points
and associated order polynomials
${\Omega^{\circ}(P_{1},x),\Omega^{\circ}(P_{2},x),\cdots,\Omega^{\circ}(P_{|P|},x)}$.
The operad $\mathcal{FP}$ assign to these order polynomials the element
$\Omega^{\circ}(P(P_{1},\cdots,P_{|P|}),x)$, which corresponds to the
lexicographic sum of the posets $P_{i}$ induced by $P$. This assignment
satisfies all the necessary conditions of an operadic morphism, establishing
the desired result. ∎

The concept of Lexicographic sum is a well-established topic in order theory,
with numerous studies on its properties and applications. For instance,
Hiraguchi’s relation provides a useful formula for calculating the dimension
of the resulting poset:

$dim(P(P_{1},\cdots,P_{|P|}))=\max\\{dim(P),dim(P_{1})\cdots,dim(P_{|P|})\\},$

where $dim$ denotes the dimension of a poset. However, instead of focusing on
the specific properties of individual posets, our approach aims to identify
common properties and structures shared among different poset algebras.

###  2.1. Order series

There are two families of polynomials indexed by posets, strict order
polynomials $\Omega^{\circ}(P,x)=\sum d_{P,i}{x\choose i},$ and weak order
polynomials
$\Omega(P,x)=\sum(-1)^{|P|-i}d_{P,i}\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{x}{i}\right)\kern-3.00003pt\right)$.

The generating series associated with each of these order polynomials can be
defined. The strict order series is given by
$\mathfrak{Z}\left(P\right)=\sum_{n=1}\Omega^{\circ}(P,n)x^{n}$, and the weak
order series is defined as
$\mathfrak{Z}^{+}\left(P\right)=\sum_{n=1}\Omega(P,n)x^{n}$. To make the
notation more concise, we will use $\mathfrak{Z}\left(n\right)$ and
$\mathfrak{Z}^{+}\left(n\right)$ to denote $\mathfrak{Z}\left(\langle
n\rangle\right)$ and $\mathfrak{Z}^{+}\left(\langle n\rangle\right)$,
respectively.

In practice, the use of order series simplifies the computations and clarifies
identities. In fact, the study of order series has led to the discovery of new
binomial identities and generalizations of the Chu-Vandermonde identity, as
shown in [3]. Therefore, in this section, we focus on exploring the properties
of order series.

######  Remark 2.5.

Equation (3) reveals that the generating series of
$\Omega^{\circ}(P,x)=\sum_{j=1}^{|P|}d_{P,j}{x\choose j}$ can be expressed as

(2) $\displaystyle\mathfrak{Z}\left(P\right)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\Omega^{\circ}(P,i)x^{i}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\sum_{j=1}^{|P|}d_{P,j}{i\choose j}x^{i}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{|P|}d_{P,j}\sum_{n=1}^{\infty}{i\choose j}x^{i}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{|P|}d_{P,j}\mathfrak{Z}\left(i\right),$

and similarly, for
$\Omega(P,x)=\sum(-1)^{|P|-i}d_{P,i}\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{x}{i}\right)\kern-3.00003pt\right)$,
we have
$\mathfrak{Z}^{+}\left(P\right)=\sum(-1)^{|P|-i}d_{P,i}\mathfrak{Z}^{+}\left(i\right)$.
We refer to the basis $\\{\mathfrak{Z}\left(n\right)\\}_{n\in\mathbb{N}}$ (or
equivalently $\\{\mathfrak{Z}^{+}\left(n\right)\\}_{n\in\mathbb{N}}$) as the
inclusion-exclusion basis for strict (resp. weak) order series. It is worth
noting that while the inclusion-exclusion vectors $d_{i,P}$ were known to
Stanley, to the best of our knowledge, the only reference where they are used
to study order series/ Ehrhart series is the paper [3]. This could be
attributed to the fact that order polytopes possess favorable properties that
we cannot expect in arbitrary integer polytopes.

There is always a non-strict morphism from a poset to $\langle 1\rangle$ that
labels every element of the poset with the constant value $1$. Then, the
series $\mathfrak{Z}^{+}\left(P\right)$ starts at index $n=1$. For the strict
order series $\mathfrak{Z}\left(P\right)$, the lowest non-zero value is the
length of a maximal chain on the poset.

In category theory, the Yoneda Lemma states that an object $c$ can be fully
characterized by the functor $\hom(c,\\_)$, which captures how the object
interacts with its environment. In combinatorics, we similarly represent a
poset $P$ by its order series $\mathfrak{Z}\left(P\right)=\sum\\#Hom(P,\langle
n\rangle)x^{n}$. However, this series loses some information about the poset:
it only considers morphisms from $P$ to chains and only keeps track of the
number of such maps. Consequently, two posets can have the same order series,
and such pairs are called Doppelgängers [12, 25]. For example, we have
$\mathfrak{Z}\left(\\{w<x,w<y,w<z\\}\right)=\mathfrak{Z}\left(\\{x<y,w<z\\}\right)$.
In Lemma 2.6, we show that the operad $\mathcal{FP}$, which acts on posets,
also acts on order series, bringing all the combinatorial structure to order
series despite the information lost.

An operadic morphism $\varrho$ between $A$ and $B$ algebras over
$\mathcal{FP}$, is a map making the following diagram commutative:

${\mathcal{FP}}$${End_{A}}$${End_{B}}$$\scriptstyle{\varphi_{B}}$$\scriptstyle{\varphi_{A}}$$\scriptstyle{\varrho}$

######  Lemma 2.6.

The operad $\mathcal{FP}$ acts on strict (weak) order series. The assignments

(3) $\displaystyle u:{x\choose n}$ $\displaystyle\mapsto$
$\displaystyle\mathfrak{Z}\left(n\right)$ $\displaystyle=$
$\displaystyle\sum_{r=n}^{\infty}{r\choose n}x^{r}$ $\displaystyle=$
$\displaystyle\frac{x^{n}}{(1-x)^{n+1}},$ (4) $\displaystyle
v:\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{x}{n}\right)\kern-3.00003pt\right)$
$\displaystyle\mapsto$ $\displaystyle\mathfrak{Z}^{+}\left(n\right)$
$\displaystyle=$
$\displaystyle\sum_{r=1}^{\infty}\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{r}{n}\right)\kern-3.00003pt\right)x^{r}$
$\displaystyle=$ $\displaystyle\frac{x}{(1-x)^{n+1}},$

extend to an operadic isomorphism between algebras over the operad
$\mathcal{FP}$.

###### Proof.

We first extend the map $u$ linearly to obtain

$u\Big{(}\sum_{i=1}^{|P|}d_{P,i}{x\choose
i}\Big{)}=\sum_{i=1}^{|P|}d_{P,i}\mathfrak{Z}\left(i\right).$

Given a poset $P$ and $P_{1},\ldots,P_{|P|}$ finite posets, we define the
action of $\mathcal{FP}$ on order series by

$\displaystyle\varphi_{s}(P,\mathfrak{Z}\left(P_{1}\right),\ldots,\mathfrak{Z}\left(P_{|P|}\right))$
$\displaystyle=\sum_{i=1}^{\infty}\varphi_{p}(P,\Omega^{\circ}(P_{1},y),\ldots,\Omega^{\circ}(P_{|P|},y)|_{y=i})x^{i}.$

If we replace $P$ by $P(P_{1},\ldots,P{|P|})$ in Equation (2), we obtain the
explicit description:

$\varphi_{s}(P,\mathfrak{Z}\left(P_{1}\right),\ldots,\mathfrak{Z}\left(P_{|P|}\right))=\sum_{j=1}^{|P(P_{1},\ldots,P_{|P|})|}d_{P(P_{1},\ldots,P_{|P|}),j}\mathfrak{Z}\left(i\right).$

Because the set $\\{{x\choose i}\\}_{i\in\mathbb{N}}$ is linearly independent
over $\mathbb{Q}$, the map $u$ is a well-defined linear transformation.
Moreover, because the set $\\{\frac{x^{i}}{(1-x)^{i+1}}\\}_{i\in\mathbb{N}}$
is linearly independent over $\mathbb{Q}$, if we have $\sum a_{i}u({x\choose
i})=0$, then $a_{i}=0$, i.e., $u$ is injective and an isomorphism over its
image. It follows that the action of the operad $\mathcal{FP}$ on strict
(weak) order series is well-defined and isomorphic to the image of another
algebra over $\mathcal{FP}$. The same argument applies to weak order series. ∎

We can leverage analytic methods in our study of posets via power series. For
instance, given a poset $P$, we define the function $i(x)=\frac{1}{x}$ and the
map
$\iota(\mathfrak{Z}^{+}\left(P\right))=(-1)^{|P|+1}(\mathfrak{Z}^{+}\left(P\right)\circ{i})$.
Note that we need to know that the function $\mathfrak{Z}^{+}\left(P\right)$
is generated by the poset $P$ in order to evaluate $\iota$.

######  Lemma 2.7.

The map $\iota$ is an involution between the set of weak order series and the
set of strict order series.

###### Proof.

Let $\mathfrak{Z}\left(P\right)$ be a strict order series. Then, we have

$\displaystyle\iota(\mathfrak{Z}\left(P\right))$ $\displaystyle=$
$\displaystyle(-1)^{|P|+1}(\mathfrak{Z}\left(P\right)\circ{i})$
$\displaystyle=$ $\displaystyle(-1)^{|P|+1}(\sum
d_{P,i}\frac{x^{i}}{(1-x)^{i+1}})\circ{i}$ $\displaystyle=$
$\displaystyle(-1)^{|P|+1}(\sum(-1)^{i+1}d_{P,i}\frac{x}{(1-x)^{i+1}})$
$\displaystyle=$ $\displaystyle\sum(-1)^{|P|-i}d_{P,i}\frac{x}{(1-x)^{i+1}}$
$\displaystyle=$ $\displaystyle\mathfrak{Z}^{+}\left(P\right),$

where the second equality follows from substituting $x\rightarrow\frac{1}{x}$
and using the fact that $d_{P,i}=0$ for $i>|P|$. Thus, $\iota$ maps strict
order series to weak order series.

Since $\iota$ is its own inverse, it is an involution between the set of weak
order series and the set of strict order series. ∎

######  Lemma 2.8.

The morphism $\iota$ is an operadic involution between weak order series and
strict order series.

###### Proof.

Let $P_{1},\cdots,P_{|P|}$ be finite posets. Now, to see that $\iota$ commutes
with the operadic action, evaluate on $P(P_{1},\cdots,P_{|P|})$ to obtain:

$\displaystyle\iota(\varphi(P,\mathfrak{Z}\left(P_{1}\right),\cdots,\mathfrak{Z}\left(P_{|P|}\right)))$
$\displaystyle=$
$\displaystyle\iota(\mathfrak{Z}\left(P(P_{1},\cdots,P_{|P|})\right))$
$\displaystyle=$
$\displaystyle\mathfrak{Z}^{+}\left(P(P_{1},\cdots,P_{|P|})\right)$
$\displaystyle=$
$\displaystyle\varphi(P,\mathfrak{Z}^{+}\left(P_{1}\right),\cdots,\mathfrak{Z}^{+}\left(P_{|P|}\right))$
$\displaystyle=$
$\displaystyle\varphi(P,\iota(\mathfrak{Z}\left(P_{1}\right)),\cdots,\iota(\mathfrak{Z}\left(P_{|P|}\right))).$

The verification of
$\iota(\varphi(P,\mathfrak{Z}^{+}\left(P_{1}\right),\cdots,\mathfrak{Z}^{+}\left(P_{|P|}\right)))=\varphi(P,\iota(\mathfrak{Z}^{+}\left(P_{1}\right)),\cdots,\iota(\mathfrak{Z}^{+}\left(P_{|P|}\right)))$
follows the same reasoning. ∎

######  Remark 2.9.

The series $\mathfrak{Z}\left(\varnothing\right)=\frac{1}{(1-x)}$ is the unit
of strict order series under the action of the operad $\mathcal{FP}$. The
image of this unit under the involution $\iota$, denoted by
$\mathfrak{Z}^{+}\left(\mayadigit{0}\right):=(-1)^{1}(\mathfrak{Z}^{+}\left(\varnothing\right)\circ\iota)=\frac{x}{1-x}$,
it is the unit of weak order series under the action of the operad
$\mathcal{FP}$. We introduce the symbol $\mayadigit{0}$ to represent an
expansion around infinity of the power series
$\mathfrak{Z}\left(\varnothing\right)$.

####  2.1.1. Explicit Description of Low-Level Operations

We now describe the action of a poset $P$ on order series for small $P$. We
denote the Hadamard product of two power series $f(x)$ and $g(x)$ as
$f(x)\sqcup g(x)$. The table in Figure 2 provides an explicit description of
the action for series-parallel posets. For more information, please refer to
[3].

Poset |  Evaluation on order series
---|---
$\\{x,y\\}$ |  $\\{x,y\\}(\mathfrak{Z}\left(P_{1}\right),\mathfrak{Z}\left(P_{2}\right))=\mathfrak{Z}\left(P_{1}\right)\sqcup\mathfrak{Z}\left(P_{2}\right)$
$\\{x<y\\}$ |  $\\{x<y\\}(\mathfrak{Z}\left(P_{1}\right),\mathfrak{Z}\left(P_{2}\right))$
|  = $\mathfrak{Z}\left(P_{1}\right)(1-x)\mathfrak{Z}\left(P_{2}\right)$
$\\{x_{1}<x_{2},x_{1}<x_{3}\\}$ |  $\\{x_{1}<x_{2},x_{1}<x_{3}\\}$ $(\mathfrak{Z}\left(P_{1}\right),\mathfrak{Z}\left(P_{1}\right),\mathfrak{Z}\left(P_{1}\right))$
|
$=\mathfrak{Z}\left(P_{1}\right)(1-x)(\mathfrak{Z}\left(P_{2}\right)\sqcup\mathfrak{Z}\left(P_{3}\right))$
Figure 2. Operations on power series parametrized by posets.

Note that

$\displaystyle\\{x<y\\}(\mathfrak{Z}\left(k\right),\mathfrak{Z}\left(j\right))$
$\displaystyle=$
$\displaystyle\mathfrak{Z}\left(k\right)(1-x)\mathfrak{Z}\left(j\right)$
$\displaystyle=$ $\displaystyle\mathfrak{Z}\left(k+j\right),$

as expected. This is because the ordinal sum of posets satisfies
$\\{x<y\\}(\langle k\rangle,\langle j\rangle)=\langle k+j\rangle$.

Consider the following identity [23, (6.44)],[3]:

(5) ${x\choose n}{x\choose s}=\sum_{j=0}^{s}{n+j\choose s}{s\choose
j}{x\choose n+j},$

where $s\leq n$.

According to [3], the Hadamard product of the order series of chains can be
computed using the following formula:

(6)
$\mathfrak{Z}\left(n\right)\sqcup\mathfrak{Z}\left(s\right)=\sum_{j=0}^{s}{n+j\choose
s}{s\choose j}\mathfrak{Z}\left(n+j\right),$

where $s\leq n$. This result can be proven using the fact that $Poly(P\sqcup
Q)$ is the Minkowski sum of $Poly(P)$ and $Poly(Q)$, and then we count the
simplices on the cannonical triangulation of $Poly(P\sqcup Q)$.

######  Example 2.10.

We define the action of the operad of finite posets on natural numbers by
considering the action of the operad on order series, where we record only the
smallest chain after a poset acts on order series. For example, using Equation
(6), we define $\\{x,y\\}(m,n)=\max\\{m,n\\}$, and using

$\\{1<2\\}(\mathfrak{Z}\left(m\right),\mathfrak{Z}\left(n\right))=\mathfrak{Z}\left(m+n\right),$

we define $\\{1<2\\}(m,n)=m+n$. By extending these definitions to all real
numbers, we recover the tropical numbers [42] $(\mathbb{R},+,\max,-\infty)$.

One could wonder about the actions of higher poset operations. For example, we
compute $\\{x<y>z<w\\}(m,n,r,s)=\max(m+n,n+r,r+s)$. Strictly speaking, if we
restrict to the operad generated by $\\{x,y\\}$ and $\\{x<y\\}$, then this is
not a $4$-ary operation, but the result of evaluating a $6$-ary operation on
duplicated inputs. This structure could be explained better with a product and
permutation category (PROP), but we will not study such an approach in this
paper.

In general, given a poset $P$, evaluating on $(n_{1},\cdots,n_{|P|})$ returns
the length of the maximal chain in the lexicographic sum $P(\langle
n_{1}\rangle,\cdots,\langle n_{|P|}\rangle)$. Then, every poset operation on
the tropical natural numbers is obtained by evaluating an operation involving
only $\\{1<2\\}$ and $\\{1,2\\}$, perhaps on duplicated inputs.

######  Lemma 2.11.

The operad of finite posets acting on order series contains a $4$-ary
operation that is associative and non-commutative, and cannot be generated by
only using the binary operations $\\{x<y\\}$ and $\\{x,y\\}$.

###### Proof.

We used Mathematica [52] to compute the values of Table 3, as explained in the
notebook [19]. Table 3 contains the 4th-ary operations generated by
$\\{x,y\\}$ and $\\{x<y\\}$, evaluated on
$(\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right))$.

Poset |  Vector
---|---
$\\{x<y<z<w\\}$ |  $\mathfrak{Z}\left(4\right)$
$\\{x<y<z,w\\}$ |  $3\mathfrak{Z}\left(3\right)+4\mathfrak{Z}\left(4\right)$
$\\{x<y,z<w\\},$ $\\{x<y,x<z,x<w\\},$ $\\{y<x,z<x,w<x\\}$ |  $\mathfrak{Z}\left(2\right)+6\mathfrak{Z}\left(3\right)+6\mathfrak{Z}\left(4\right)$
$\\{x,y,z<w\\}$ |  $4\mathfrak{Z}\left(2\right)+15\mathfrak{Z}\left(3\right)+12\mathfrak{Z}\left(4\right)$
$\\{x,y,z,w\\}$ |  $\mathfrak{Z}\left(1\right)+14\mathfrak{Z}\left(2\right)+36\mathfrak{Z}\left(3\right)+24\mathfrak{Z}\left(4\right)$
$\\{x<y>z,w\\},$ $\\{x,y>z<w\\}$ |  $2\mathfrak{Z}\left(2\right)+9\mathfrak{Z}\left(3\right)+8\mathfrak{Z}\left(4\right)$
$\\{x<y>z<w\\}$ |  $\mathfrak{Z}\left(2\right)+5\mathfrak{Z}\left(3\right)+5\mathfrak{Z}\left(4\right)$
Figure 3. Vectors of the evaluation of the 4th-ary operation on
$(\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right))$

While $\\{x<y\\}$ and $\\{x,y\\}$ are binary, commutative, associative
operations, Table 3 shows that the operation $\\{x<y>z<w\\}$ is a non-
commutative $4^{th}$-ary operation that is not generated by $\\{x<y\\}$ and
$\\{x,y\\}$. We know that this operation is non-commutative because

(7)
$\displaystyle\\{x<y>z<w\\}(\mathfrak{Z}\left(2\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right))$
$\displaystyle=$
$\displaystyle\\{x<y>z<w\\}(\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(2\right),\mathfrak{Z}\left(1\right))$
$\displaystyle=$ $\displaystyle
9\mathfrak{Z}\left(5\right)+11\mathfrak{Z}\left(4\right)+3\mathfrak{Z}\left(2\right),$
(8)
$\displaystyle\\{x<y>z<w\\}(\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(2\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right))$
$\displaystyle=$
$\displaystyle\\{x<y>z<w\\}(\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(2\right))$
$\displaystyle=$ $\displaystyle
7\mathfrak{Z}\left(5\right)+8\mathfrak{Z}\left(4\right)+2\mathfrak{Z}\left(2\right).$

From Example 2.10, one can wonder if there is an $n$-ary operation $f$
evaluated on duplicated inputs that is generated by $\\{x,y\\},\\{x<y\\}$ and
sends
$(\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right),\cdots)$
to
$\mathfrak{Z}\left(2\right)+5\mathfrak{Z}\left(3\right)+5\mathfrak{Z}\left(4\right)$.
If such an operation exists, there is a series parallel poset $P_{f}$
associated with it. A series parallel poset is a poset generated by
$\\{x,y\\},\\{x<y\\}$.

From [3, Proposition 3.3], parts 2) and 3) it follows that the maximal chain
of $P_{f}$ has length 2 and the poset $P_{f}$ has 4 points. This rules out the
possibility of having an operation with a higher number of inputs returning
$\mathfrak{Z}\left(2\right)+5\mathfrak{Z}\left(3\right)+5\mathfrak{Z}\left(4\right)$.
∎

Note that $\\{x<y,x<z\\}$ is also non-commutative since the value of $x$
cannot be exchanged with the value of $y$ or $z$. However, $\\{x<y,x<z\\}$ can
be expressed as the composition (on $w$) of the binary operations $\\{y,z\\}$
and $\\{x<w\\}$.

Some know properties of $\\{x<y>z<w\\}$ are:

$\displaystyle\\{x<y>z<w\\}(\mathfrak{Z}\left(a\right),\mathfrak{Z}\left(b\right),\mathfrak{Z}\left(c\right),\mathfrak{Z}\left(d\right))$
$\displaystyle=$
$\displaystyle\\{x<y>z<w\\}(\mathfrak{Z}\left(d\right),\mathfrak{Z}\left(c\right),\mathfrak{Z}\left(b\right),\mathfrak{Z}\left(a\right))$
$\displaystyle\\{x<y>z<w\\}(\mathfrak{Z}\left(k\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right))$
$\displaystyle=$
$\displaystyle(\frac{(k+3-2)(k+3+1)}{2})\mathfrak{Z}\left(k+3\right)$
$\displaystyle+({k+2\choose
2}+\frac{(k+2-2)(k+2+1)}{2})\mathfrak{Z}\left(k+2\right)$
$\displaystyle+{k+1\choose 2}\mathfrak{Z}\left(k+1\right)$
$\displaystyle\\{x<y>z<w\\}(\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(k\right),\mathfrak{Z}\left(1\right),\mathfrak{Z}\left(1\right))$
$\displaystyle=$
$\displaystyle(2(k+3)-3)\mathfrak{Z}\left(k+3\right)+(2(k+2)-3)\mathfrak{Z}\left(k+2\right)$
$\displaystyle+[(k+2-1)]\mathfrak{Z}\left(k+2\right)+(k+1-1)\mathfrak{Z}\left(k+1\right)$
$\displaystyle\\{x<y>z<w\\}(\mathfrak{Z}\left(\varnothing\right),\mathfrak{Z}\left(b\right),\mathfrak{Z}\left(c\right),\mathfrak{Z}\left(d\right))$
$\displaystyle=$
$\displaystyle\mathfrak{Z}\left(c\right)(1-x)(\mathfrak{Z}\left(b\right)\sqcup\mathfrak{Z}\left(d\right))$
$\displaystyle\\{x<y>z<w\\}(\mathfrak{Z}\left(a\right),\mathfrak{Z}\left(\varnothing\right),\mathfrak{Z}\left(c\right),\mathfrak{Z}\left(d\right))$
$\displaystyle=$
$\displaystyle\mathfrak{Z}\left(a\right)\sqcup\mathfrak{Z}\left(c+d\right)$
$\displaystyle\\{x<y>z<w\\}(\mathfrak{Z}\left(a\right),\mathfrak{Z}\left(b\right),\mathfrak{Z}\left(\varnothing\right),\mathfrak{Z}\left(d\right))$
$\displaystyle=$
$\displaystyle\mathfrak{Z}\left(a+b\right)\sqcup\mathfrak{Z}\left(d\right)$
$\displaystyle\\{x<y>z<w\\}(\mathfrak{Z}\left(a\right),\mathfrak{Z}\left(b\right),\mathfrak{Z}\left(c\right),\mathfrak{Z}\left(\varnothing\right))$
$\displaystyle=$
$\displaystyle\mathfrak{Z}\left(b\right)(1-x)(\mathfrak{Z}\left(a\right)\sqcup\mathfrak{Z}\left(c\right))$

##  3\. The union of points

In this section we study the case in which the poset is the disjoint union of
points. As a result, we obtain conceptual proofs of several results on
Eulerian numbers.

Eulerian polynomials have been extensively studied and have numerous
applications [16, 20, 38]. In this work, we adopt the modern definition of
Eulerian polynomials, which is given by the following generating function:

(9) $\sum_{n=0}^{\infty}A_{n}(t)\frac{x^{n}}{n!}=\frac{t-1}{t-e^{(t-1)x}}.$

By multiplying Equation (9) by $x$ and substituting $x=\frac{1}{1-t}$, we
obtain

(10) $\sum_{n=0}^{\infty}\frac{A_{n}(t)}{(1-t)^{n+1}n!}=\frac{1}{e^{-1}-t}.$

It is well known that the Eulerian polynomials satisfy the identity:

(11)
$x\frac{A_{n}(x)}{(1-x)^{n+1}}=\sum_{k=1}^{\infty}k^{n}x^{k}=\sqcup_{n}\mathfrak{Z}\left(1\right).$

Here, we realize the right side of the equation as the strict order series of
the disjoint union of $n$ points. We consider the $n$-cube as the order
polytope of the disjoint union of $n$ points.

Given a polytope $Po$, we can define $L(Po,t)$ as the count of lattice points
in the expansion of $tPo$ for a given value of $t$. By summing $L(Po,t)$ over
all positive integers $t$, we arrive at the Ehrhart series of $Po$, which can
be expressed as
$\sum_{t=1}^{\infty}L(Po,t)x^{t}=\frac{h^{\ast}(x)}{(1-x)^{d+1}}$, where
$h^{\ast}(x)$ is a polynomial of degree less than $d$. To begin our
exploration, let us consider a poset $P$. By computing the Ehrhart series of
the order polytope of $P$, we can then utilize Equation (12) to deduce the
strict order series of $P$:

(12) $\mathfrak{Z}^{+}\left(P\right)=x\frac{h^{*}(x)}{(1-x)^{d+1}}.$

When $P$ is the disjoint union of $n$ points, then the $n$-cube serves as the
order polytope of $P$.

######  Proposition 3.1.

For a poset $P$, the Ehrhart series of the order polytope of $P$
$\frac{h^{*}(x)}{(1-x)^{\|P\|+1}}$ satisfies $h^{*}_{\|P\|}=0$.

###### Proof.

Note that $\frac{h_{\|P\|}^{*}x^{\|P\|+1}}{(1-x)^{\|P\|+1}}$ cannot be in the
algebra $\\{\mathfrak{Z}^{+}\left(i\right)=\frac{x}{(1-x)^{i+1}}\\}_{1\leq
i\leq\|P\|}$ unless $h_{\|P\|}^{*}=0$. ∎

By utilizing Equation (11), we can see that the $h^{*}(x)$ polynomial of the
$n$-cube is, in fact, the $n-$Eulerian polynomial
$A_{n}(x)={\sum_{m=0}^{n-1}A(n,m)}x^{m}$.

Further applying $x=-1,t=-1$ and $\psi_{n}(t):=A_{n}(-t)$ to Equation (11), we
can deduce the following entry from [6, Chapter 5, Entry 4]: For $|P|<1$ and
$n\geq 0$,

$(p+1)^{-n-1}\psi_{n}(p)=\sum_{k=0}^{\infty}(k+1)^{n}(-p)^{k}.$

######  Example 3.2.

By combining Lemma 2.3 and Remark 2.5, we can express the order series of the
union of $n$ points as:

(13) $\sqcup_{n}\mathfrak{Z}\left(1\right)=\sum_{i=1}^{n}d_{\sqcup_{n}\langle
1\rangle,i}\mathfrak{Z}\left(i\right),$

where $d_{\sqcup_{n}\langle 1\rangle,i}$ is the number of internal $i$-faces
in the canonical triangulation of the $n$-cube that are obtained as the
intersection of $n-i+1$ maximal $n$-simplices.

######  Corollary 3.3.

For a fixed integer $n$, there exists a sequence of natural numbers
${d_{\sqcup_{n}\langle 1\rangle,i}}{1\leq i\leq n}$ such that for all $r>0$,

  * •

$r^{n}=\sum_{i=1}^{n}d_{\sqcup_{n}\langle 1\rangle,i}{r\choose i}$,

where ${q\choose p}=0$ if $q<p$.

###### Proof.

By using Equation (3) on Equation (13) and comparing the coefficients with
Equation (11), the result follows. ∎

This result is a significant improvement over Ramanujan’s result, which is
limited to the case where $1\leq r\leq n$ [6, Chapter 5, Corollary 6, i)]. The
numbers $d_{\sqcup_{n}\langle 1\rangle,i}$ are closely related to the Stirling
numbers of the second kind $S(n,i)$, which count the number of non-empty
partitions of a set with $n$ labeled objects. Specifically, we have
$d_{\sqcup_{n}\langle 1\rangle,i}=i!S(n,i)$ [33, Theorem 5.6]. The values of
$d_{\sqcup_{n}\langle 1\rangle,k}$ for $1\leq k\leq n$ are explicitly known:
for $n=2$, they are ${1,2}$, and for $n=3$, they are ${1,6,6}$, which
correspond to the sequence A019538 in the Online Encyclopedia of Integer
Sequences [26].

In the following proposition we use our poset point of view to provide a
conceptual proof the equality of two power series.

######  Proposition 3.4.

For any $n\in\mathbb{N}$, we have:

(14)
$\sqcup_{n}\mathfrak{Z}^{+}\left(1\right)=\sqcup_{n}\mathfrak{Z}\left(1\right).$

###### Proof.

A disjoint union of points cannot distinguish between strict and non-strict
morphisms. Therefore, the weak and the strict order series are the same. In
other words, for the union of points, every weak order-preserving map is also
a strict order-preserving map. ∎

######  Corollary 3.5.

Let $P=\sqcup_{n}\langle 1\rangle$, then

$\sum_{k=1}^{n}k!S(n,k)x^{k-1}(1-x)^{n-k}=\sum_{k=1}^{n}(n-k)!S(n,n-k)(1-x)^{k}.$

###### Proof.

Starting from $\mathfrak{Z}\left(P\right)=\mathfrak{Z}^{+}\left(P\right)$ and
using Lemma 2.3, we obtain:

$\sum_{k=1}^{n}\frac{k!S(n,k)x^{k}}{(1-x)^{k+1}}=\sum_{k=1}^{n}(-1)^{k}\frac{n-k!S(n,n-k)x}{(1-x)^{n-k+1}},$

Multiplying both sides by $\frac{(1-x)^{n+1}}{x}$ yields the desired identity.
∎

Consider Ramanujan’s notation
$\psi_{n}(p)=\sum_{k=0}^{n-1}F_{k+1}(n)(-p)^{k}$. We are now ready to prove
the following statement from [6, Chapter 5, Entry 6]:

######  Entry 3.6.

Let $1\leq r\leq n.$ Then

  1. (1)

$F_{r}(n)=F_{n-r+1}(n)$,

  2. (2)

$r^{n}=\sum_{k=0}^{r-1}F_{r-k}(n){n+k\choose k},$

  3. (3)

$F_{r}(n)=\sum_{k=0}^{r-1}(-1)^{k}{n+1\choose k}(r-k)^{n}.$

The identity (2) is due to Worpitzky [53] while the identity (3) is due to
Euler [20]. Remember that the $h^{*}$ polynomial of the $n$–cube
$\sqcup_{n}\mathfrak{Z}^{+}\left(1\right)=x\frac{h^{*}(x)}{(1-x)^{n+1}}$ is
the Eulerian polynomial $\sum_{i=0}^{n-1}A(n,i)x^{i}$.

######  Corollary 3.7.

Let $1\leq r\leq n$. Then

  1. (1)

$A(n,i)=A(n,n-1-i).$

  2. (2)

$k^{n}=\sum_{i=0}^{n-1}A(n,i){k+i\choose
n},k^{n}=\sum_{i=0}^{n-1}A(n,i)\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{k-i}{n}\right)\kern-3.00003pt\right)$.

  3. (3)

$A(n,i)=\sum_{r=0}^{n+1}(-1)^{r}{n+1\choose r}(i+1-r)^{n}.$

###### Proof.

$i)$ The right-hand side of Equation (14) is related to the left-hand side via
the Lemma 2.7. We then replace Equation (12) to obtain:

$\displaystyle x\frac{\sum_{i=0}^{n-1}A(n,i)x^{i}}{(1-x)^{n+1}}$
$\displaystyle=$
$\displaystyle(-1)^{n+1}\frac{1}{x}\frac{\sum_{i=0}^{n-1}A(n,i)\frac{1}{x^{i}}}{(1-\frac{1}{x})^{n+1}}$
$\displaystyle=$
$\displaystyle\frac{\sum_{i=0}^{n-1}A(n,i){x}^{n-i}}{(1-{x})^{n+1}}.$

Then $\ref{ref-i})$ follows by comparing the $h^{*}$ vectors degree wise.
$ii)$ Worpitzky identity follows from comparing degree-wise the coefficients
of
$\sum_{k=1}^{\infty}k^{n}x^{k}=\frac{\sum_{i=0}^{n-1}A(n,i)x^{n-i}}{(1-x)^{n+1}}.$
The multiset version follows from
$\sum_{k=1}^{\infty}k^{n}x^{k}=x\frac{\sum_{i=0}^{n-1}A(n,i)x^{i}}{(1-x)^{n+1}},$
but due to the symmetry of the coefficients, it is the same identity. $iii)$
Follows the same technique as in [6]. ∎

##  4\. Poset version of Ramanujan’s results

In this section we show that certain results of Ramanujan are parametrized by
disjoint union of posets. We say that we obtained a poset version of Ramanujan
results when we prove these results for arbitrary finite posets.

Ramanujan gives the following examples:

######  Entry 4.1 ([6, Chapter 5, Example 1,2.]).

$\sum_{k=1}^{\infty}\frac{k^{5}}{2^{k}}=1082,\sum_{k=1}^{\infty}\frac{k^{5}}{3^{k}}=273/4.$

They can be justified by using $P=\sqcup^{5}\langle 1\rangle$ on the next
lemma:

######  Lemma 4.2.

Let $P$ be a finite poset and
$\mathfrak{Z}\left(P\right)=\sum_{i=1}^{\|P\|}d_{P,i}\mathfrak{Z}\left(i\right)$.
Let $r>1$, and $r_{0}$ the length of a maximal chain in $P$, then:

(15)
$\displaystyle\sum_{n=r_{0}}^{\infty}\frac{\Omega^{\circ}(P,n)}{r^{n}}=\sum_{i=1}^{\|P\|}(-1)^{i+1}d_{P,i}\frac{r}{(1-r)^{i+1}}=(-1)^{\|P\|+1}\mathfrak{Z}^{+}\left(P\right),$
(16)
$\displaystyle\sum_{n=1}^{\infty}\frac{\Omega(P,n)}{r^{n}}=(-1)^{\|P\|+1}\sum_{i=1}^{\|P\|}d_{P,i}\frac{r^{i}}{(1-r)^{i+1}}=(-1)^{\|P\|+1}\mathfrak{Z}\left(P\right).$

###### Proof.

From Lemma 2.3 we obtain the equality

$(-1)^{\|P\|+1}\sum_{k=r_{0}}{\Omega^{\circ}(P,k)}{x^{k}}=(-1)^{\|P\|+1}\sum_{i=1}^{\|P\|}d_{P,i}\frac{x^{i}}{(1-x)^{i+1}}.$

Note that $\sum\frac{k^{\|P\|}}{x^{k}}$ converges on $|x|>1$, We now apply
$\iota$ (from Lemma 2.7) on both sides to obtain

$\sum_{k=r_{0}}\frac{\Omega^{\circ}(P,k)}{x^{k}}=\sum_{i=1}^{\|P\|}(-1)^{i+1}d_{P,i}\frac{x}{(1-x)^{i+1}}$

This proves (15). The identity (16) follows from the same reasoning. ∎

######  Example 4.3.

For $P=\langle 1\rangle*(\langle 1\rangle\sqcup\langle 1\rangle\sqcup\langle
1\rangle),$ we calculated
$\mathfrak{Z}^{+}\left(P\right)=\mathfrak{Z}^{+}\left(2\right)-6\mathfrak{Z}^{+}\left(3\right)+6\mathfrak{Z}^{+}\left(4\right)$.
Then using Equation (4) we compute

$\Omega^{\circ}(P,k)={k\choose 2}+6{k\choose 3}+6{k\choose
4}=\frac{k^{4}-2k^{3}+k^{2}}{4}.$

Evaluating Equation (15) at $r=5$, we obtain

$\displaystyle\sum_{k=2}^{\infty}\frac{k^{4}-2k^{3}+k^{2}}{4(5^{k})}$
$\displaystyle=$
$\displaystyle-\frac{5}{(1-5)^{3}}+6\frac{5}{(1-5)^{4}}-6\frac{5}{(1-5)^{5}}$
$\displaystyle=$ $\displaystyle\frac{115}{512},$

similarly,

$\Omega(P,k)={k+1\choose 2}-6{k+2\choose 3}+6{k+3\choose
4}=\frac{k^{2}(k+1)^{2}}{4},$
$\displaystyle\sum_{k=1}^{\infty}\frac{k^{2}(k+1)^{2}}{4*5^{k}}$
$\displaystyle=$
$\displaystyle-\frac{5^{2}}{(1-5)^{3}}+6\frac{5^{3}}{(1-5)^{4}}-6\frac{5^{4}}{(1-5)^{5}}$
$\displaystyle=$ $\displaystyle\frac{575}{512}.$

Consider the Riemann zeta function
$\zeta(i)=\sum_{n=1}^{\infty}\frac{1}{n^{i}}$, whose values on even natural
numbers are well-known. The Bernoulli numbers are defined by the generating
function $\frac{x}{1-e^{x}}=\sum_{m=0}^{\infty}B_{m}\frac{x^{m}}{m!}$, and
Euler famously showed that $\zeta(2n)=-\frac{(2\pi i)^{2n}B_{2n}}{2(2n)!}$.

######  Remark 4.4.

Although the irrationality of $\zeta(3)$ is established [2], the status of the
odd zeta values remains elusive. It is known that one out of
$\zeta(5),\zeta(7),\zeta(9),\zeta(11)$ is irrational [54], and at least two
out of $\zeta(5),\zeta(7),\cdots,\zeta(35)$ are irrational [29, 40]. In fact,
an elementary proof that one out of $\zeta(5),\zeta(7),\cdots,\zeta(25)$ is
irrational is provided in [55].

Despite the lack of complete understanding of odd zeta values, remarkable
identities involving these numbers can still be established.

######  Entry 4.5 ([6, Chapter 5, Collolary 6 iv)]).

Fix $n$ natural number. Then there are $\\{A_{k}\\}_{1\leq k\leq
n}\in\mathbb{Z}$ such that

$\sum_{k=1}^{\infty}(-1)^{k+1}k^{n}(\zeta(k+1)-1)=(-1)^{n}+(-1)^{n}2^{-n-1}\psi_{n}(1)+\sum_{k=1}^{n}(-1)^{k+1}A_{k}\zeta(k+1).$

Consider $\sum_{k}(-1)^{k+1}p(k)(\zeta(k+1)-1)$ where
$p(t)=\sum_{n=1}^{N}b_{n}t^{n}$ is a polynomial with rational coefficients.
Using Ramanujan’s results we obtain

(17) $\displaystyle\sum_{k}(-1)^{k+1}p(k)(\zeta(k+1)-1)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}b_{n}\sum_{k}(-1)^{k+1}k^{n}(\zeta(k+1)-1)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{N}b^{\prime}_{i}\zeta(i+1)+b_{0}^{\prime}.$

However, we do not know much about the corresponding coefficients
$b_{i}^{\prime}$. In the following theorem we generalized Ramanujan identity
by interpreting $k^{n}$ as the orden polynomial of disjoint union of
$n$-points. Then, when the polynomial is the order polynomial of a finite
poset, we show that the coefficients of the sum in the base
$\zeta(2),\cdots,\zeta(n+1)$ are integers multiplied by alternating signs.

######  Theorem 4.6.

Fix a finite poset $P$, let $r_{0}$ the length of a maximal chain in $P$, and
let $n=\|P\|$. There are $\\{d_{P,i}\in\mathbb{N}\\}_{1\leq i\leq n}$ such
that

(18)
$\sum_{k=r_{0}}^{\infty}{(-1)^{k+1}\Omega^{\circ}(P,k)}(\zeta(k+1)-1)=\sum_{i=1}^{\|P\|}(-1)^{i+1}d_{P,i}(\zeta(i+1)-1-\frac{1}{2^{i+1}}).$

###### Proof.

We divide by $r$ on both sides of Equation (15), obtaining the Ehrhart series
of the order polytope of $P$ in the inclusion exclusion basis, we evaluate on
$-r$ to obtain:

(19)
$\displaystyle\sum_{k=1}^{\infty}(-1)^{k+1}\Omega^{\circ}(P,k)\frac{1}{r^{k+1}}=\sum_{i=1}^{|P|}(-1)^{i+1}d_{P,i}\frac{1}{(1+r)^{i+1}}.$

$\displaystyle\sum_{k=r_{0}}^{\infty}{(-1)^{k+1}\Omega^{\circ}(P,k)}(\zeta[k+1]-1)$
$\displaystyle=$
$\displaystyle\sum_{r=2}^{\infty}\sum_{k=r_{0}}^{\infty}\frac{\Omega^{\circ}(P,k)}{(-r)^{k+1}}$
$\displaystyle=$
$\displaystyle\sum_{r=2}^{\infty}\sum_{i=1}^{\|P\|}\frac{(-1)^{i+1}d_{P,i}}{(1+r)^{i+1}}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{\|P\|}(-1)^{i+1}d_{P,i}(\zeta[i+1]-1-\frac{1}{2^{i+1}}).$

∎

Consider an identity that involves only expressions of the form
$\frac{1}{x^{\alpha}},\frac{1}{(x\pm 1)^{\beta}},\cdots$, where the exponents
are natural numbers. If the identity holds for all values of $x$ greater than
or equal to a fixed $n$, and the sum of these expressions converges, then by
summing these identities, one can obtain an identity involving zeta values.
This process, referred to as expressing a relation in finite form, has been
discussed in [37]. It is worth noting that this technique can be used to
evaluate every possible subset of the real numbers, as long as the limit of
partial sums converges.

Since zeta values hold great significance, we are particularly interested in
this type of identity. The question that arises is how to discover more such
identities involving fractions $\frac{1}{x^{\alpha}},\frac{1}{(x\pm
1)^{\beta}},\cdots$. One approach is to draw from the theory of posets, as we
have done to obtain families of such relations, the Equation (18) is the
finite form expression of Equation (19).

######  Example 4.7.

For $P=\langle 1\rangle*(\langle 1\rangle\sqcup\langle 1\rangle\sqcup\langle
1\rangle)$, we obtain $d_{P,1}=0,d_{P,2}=1,d_{P,3}=6,$ and $d_{P,4}=6$:

$\displaystyle\sum_{n=2}^{\infty}(-1)^{k+1}\frac{k^{4}-2k^{3}+k^{2}}{4}(\zeta(k+1)-1)$
$\displaystyle=$
$\displaystyle-(\zeta(3)-1-\frac{1}{2^{3}})+6(\zeta(4)-1-\frac{1}{2^{4}})$
$\displaystyle-6(\zeta(5)-1-\frac{1}{2^{5}}).$

##  5\. Action of the operad of posets on zeta values

Christian Goldbach (1690-1764) showed that

$\sum_{n=2}^{\infty}(\zeta(n)-1)=1.$

In [7] series with rational coefficients and terms $\zeta(n)-1$ are called
rational zeta series, see also [1, 8, 10, 15, 22, 44].

Consider

(20) $\displaystyle\sum_{n=k}^{\infty}{n\choose k}(\zeta(n+1)-1)=\zeta(k+1),$

discovered by [7].

For sums $\sum_{i=0}a_{i}{x\choose i}$ with $a_{i}\in\mathbb{Q}$ and $a_{i}=0$
except for a finite number of terms, we define

$\displaystyle\tilde{n}(\sum_{i=0}a_{i}{x\choose i})$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}(\sum_{i=0}a_{i}{n\choose i})(\zeta(n+1)-1)$
$\displaystyle=$ $\displaystyle\sum_{i=0}a_{i}\zeta(i+1).$

From Theorem 4.6 we deduce the following identity:

(21) $\displaystyle\sum_{n=k}^{\infty}(-1)^{n+1}{n\choose k}(\zeta(n+1)-1)$
$\displaystyle=$ $\displaystyle(-1)^{k+1}(\zeta(k+1)-1-1/2^{k+1}),$

together with

$\sum_{n=1}^{\infty}(-1)^{n+1}(\zeta(n+1)-1)=1/2,$

they define a linear function $\tilde{n}_{2}$ that sends a finite sum with
rational coefficients $\sum_{i=0}a_{i}{x\choose i}$ with $a_{i}=0$ except for
a finite number of terms, into

$\displaystyle\sum_{k=1}^{\infty}(-1)^{k+1}(\sum_{i=0}a_{i}{k\choose
i})(\zeta(k+1)-1)$ $\displaystyle=$
$\displaystyle\frac{a_{0}}{2}+\sum_{i=1}^{r}(-1)^{i+1}a_{i}(\zeta(i+1)-1-\frac{1}{2^{i+1}}).$

We will use the functions $\tilde{n},\tilde{n}_{2}$ to define the action of
the operad of posets on zeta values.

From Theorem 2.4 we know there is an action of the operad of posets on the
vector space $\mathbb{Z}[\\{{x\choose i}\\}_{i\in\mathbb{N}}]$, we show how to
transfer the action to $\mathbb{Z}[\\{\tilde{n}({x\choose
i})\\}_{i\in\mathbb{N}}]$ and to $\mathbb{Z}[\\{\tilde{n}_{2}({x\choose
i})\\}_{i\in\mathbb{N}}]$. The linear map $\tilde{n}$ will become trivially an
operadic morphism.

For a poset $P$, with order series
$\Omega^{\circ}(P,x)=\sum_{i=1}^{|P|}d_{P,i}{x\choose i}$, we define
$\mathfrak{Z}_{\mathbb{N}}\left(P\right):=\tilde{n}(\Omega^{\circ}(P,x))$ and
$\mathfrak{Z}_{\mathbb{N}}^{+}\left(P\right):=(-1)^{|P|+1}\tilde{n}_{2}(\Omega^{\circ}(P,x))$.

######  Corollary 5.1.

The operad $\mathcal{FP}$ acts non-trivially on the numbers
$\\{\mathfrak{Z}_{\mathbb{N}}^{+}\left(P\right)\\}_{P\hbox{ finite poset}}$
and on $\\{\mathfrak{Z}_{\mathbb{N}}\left(P\right)\\}_{P\hbox{ finite
poset}}$.

###### Proof.

The action of a poset $P$ on
$\mathfrak{Z}_{\mathbb{N}}^{+}\left(P_{1}\right),\cdots,\mathfrak{Z}_{\mathbb{N}}^{+}\left(P_{|P|}\right)$,
is given by

$(-1)^{|P(P_{1},\cdots,P_{|P|})|+1}\tilde{n}_{2}(\Omega^{\circ}(P(P_{1},\cdots,P_{|P|}),x)).$

We see that the action is not trivial since $\Omega^{\circ}(\langle
k\rangle,x)$ is assigned to $(\zeta(k+1)-1-\frac{1}{2^{k+1}})$, and different
chains are sent to different zeta values. We see that the action of the operad
of posets is different from the action of the operad of series parallel posets
because from Lemma 2.11, the action of $\\{x<y>z<w\\}$ is not equal to the
action of any quaternary operation generated by $\\{x,y\\}$ and $\\{x<y\\}$.

The same arguments apply to $\mathfrak{Z}_{\mathbb{N}}\left(P\right)$. ∎

We describe the action of a poset in $\mathbb{Z}\\{{x\choose i}\\}$ by
computing the integer coordinates, we call these coordinates structural
constants. We use this point of view to find the action on zeta values of the
operad of finite posets. First compute the constants on polytopes or order
series, and then the map $\tilde{n},\tilde{n}_{2}$ force the constants to
appear on the zeta value side. For example, the order polytope of the
Minkowski sum of $\Delta[2]$ and $\Delta[m]$ can be written as
$\cup_{m+1}\Delta[m+1]\setminus(\cup_{m}\Delta[m])$. The action of $\\{x,y\\}$
on posets $P,Q$ is isomorphic to the action of the Minkowski sum on the
corresponding order polytopes [3] $Poly(P),Poly(Q)$. Then we use
$\tilde{n}_{2}$ to conclude that
$\\{x,y\\}(\mathfrak{Z}_{\mathbb{N}}^{+}\left(2\right),\mathfrak{Z}_{\mathbb{N}}^{+}\left(m\right))$
is equal to

$=-m(\zeta(m+1)-1-\frac{1}{2^{m+1}})+(m+1)(\zeta(m+2)-1-\frac{1}{2^{m+2}}).$

Using Equation (8) we can evaluate the associative non-commutative quaternary
operation $\\{x<y>z<w\\}$ on
$(\mathfrak{Z}_{\mathbb{N}}^{+}\left(1\right),\mathfrak{Z}_{\mathbb{N}}^{+}\left(2\right),\mathfrak{Z}_{\mathbb{N}}^{+}\left(1\right),\mathfrak{Z}_{\mathbb{N}}^{+}\left(1\right))$
and we obtain

$\displaystyle\\{x<y>z<w\\}(\mathfrak{Z}_{\mathbb{N}}^{+}\left(1\right),\mathfrak{Z}_{\mathbb{N}}^{+}\left(2\right),\mathfrak{Z}_{\mathbb{N}}^{+}\left(1\right),\mathfrak{Z}_{\mathbb{N}}^{+}\left(1\right))$
$\displaystyle=$ $\displaystyle
2\mathfrak{Z}_{\mathbb{N}}^{+}\left(2\right)-8\mathfrak{Z}_{\mathbb{N}}^{+}\left(3\right)+5\mathfrak{Z}_{\mathbb{N}}^{+}\left(4\right).$

In addition to the structural constants, the order series inherits calculus
identities that are valid in this context. For example, consider the following
proposition from [3, Proposition 4.1]:

$\displaystyle\mathfrak{Z}\left(s\right)\sqcup(\mathfrak{Z}\left(p\right)*\mathfrak{Z}\left(q\right))$
$\displaystyle=$
$\displaystyle\sum_{a+c=s}(\mathfrak{Z}\left(a\right)\sqcup\mathfrak{Z}\left(p\right))*(\mathfrak{Z}\left(c\right)\sqcup\mathfrak{Z}\left(q\right))$
$\displaystyle-$
$\displaystyle[\sum_{a+c=s-1}(\mathfrak{Z}\left(a\right)\sqcup\mathfrak{Z}\left(p\right))*(\mathfrak{Z}\left(c\right)\sqcup\mathfrak{Z}\left(q\right))]*\mathfrak{Z}\left(1\right),$

This proposition can be proven by understanding
$\sqcup\mathfrak{Z}\left(n\right)$ as a differential operator. We work with
order series instead of order polynomials because we frequently encounter
similar expressions that are better understood with the power series
structure.

At the level of zeta values, we obtain:

$\displaystyle\mathfrak{Z}_{\mathbb{N}}^{+}\left(s\right)\sqcup(\mathfrak{Z}_{\mathbb{N}}^{+}\left(p\right)*\mathfrak{Z}_{\mathbb{N}}^{+}\left(q\right))$
$\displaystyle=$
$\displaystyle\sum_{a+c=s}(\mathfrak{Z}_{\mathbb{N}}^{+}\left(a\right)\sqcup\mathfrak{Z}_{\mathbb{N}}^{+}\left(p\right))*(\mathfrak{Z}_{\mathbb{N}}^{+}\left(c\right)\sqcup\mathfrak{Z}_{\mathbb{N}}^{+}\left(q\right))$
(22) $\displaystyle-$
$\displaystyle[\sum_{a+c=s-1}(\mathfrak{Z}_{\mathbb{N}}^{+}\left(a\right)\sqcup\mathfrak{Z}_{\mathbb{N}}^{+}\left(p\right))*(\mathfrak{Z}_{\mathbb{N}}^{+}\left(c\right)\sqcup\mathfrak{Z}_{\mathbb{N}}^{+}\left(q\right))]*\mathfrak{Z}_{\mathbb{N}}^{+}\left(1\right),$

similarly for $\mathfrak{Z}_{\mathbb{N}}\left(P\right)$.

These identities are equivalent to binomial identities.

It is natural to wonder whether an arbitrary sum of the form
$b_{1}\mathfrak{Z}_{\mathbb{N}}^{+}\left(2\right)+\cdots+b_{n-1}\mathfrak{Z}_{\mathbb{N}}^{+}\left(n\right)$,
where $b_{i}$ are integer coefficients, can be expressed as
$\mathfrak{Z}_{\mathbb{N}}^{+}\left(P\right)$ for some poset $P$. An algorithm
for finding a series-parallel poset was proposed in [3, Section 3.1], with
optimizations for Wixárika posets.

Finally, we relate the injectivity of $\tilde{n}$ and $\tilde{n}_{2}$ to a
well-known open conjecture in number theory [11]:

######  Theorem 5.2.

The following statements are equivalent:

  1. (1)

The numbers $\\{1,\zeta[2],\zeta[3],\cdots\\}$ are linearly independent over
$\mathbb{Q}$.

  2. (2)

The functions $\tilde{n}$ and $\tilde{n}_{2}$ are injective.

###### Proof.

Assuming 1. Since the order polynomials are integer valued, from linear
independence we deduce that if there are two polynomials in which
$\tilde{n}(\tilde{n}_{2})$ return the same value, each coordinate
$\zeta(i+1)-1$ must have the same coefficient. Now, for different polinomials,
there is a coordinate in which the values of $\tilde{n}(\tilde{n}_{2})$
differ, otherwise the order polynomials agree on an infinite number of
integers, but this implies that the polynomials are the same. Then 2 holds.

Assuming 2. Given a linear combination with rational coefficients, $\sum
a_{i}{\zeta[i]}=0$. We can multiply by an integer to obtain integer
coefficients: $\sum b_{i}{\zeta[i]}=0$. The preimage of this element $\sum
b_{i}{x\choose i}$ belongs to the kernel of the map, which is zero. But from
$\sum b_{i}{x\choose i}=0$ we deduce that $b_{i}=0$, due to linear
independence of $\\{{x\choose i}\\}$. ∎

Regarding the first part of the previous theorem, Rivoal [39] and Ball-Rivoal
[4] established that an infinite number of odd zeta values are linearly
independent. Several recent works have made significant improvements on the
Ball-Rivoal Theorem, including [21, 28, 35, 40, 43]. See also Remark 4.4.

##  6\. Further remarks

  * •

Consider the following theorem:

######  Theorem 6.1.

If $n$ is a natural number, then

$\sum_{n=1}^{\infty}\frac{1}{n^{k}(n+1)^{k}}=\sum^{k}_{n=0,\\\ n\neq
k-1}(1+(-1)^{k-n})\zeta({k-n}){-k\choose n}.$

This result can be found in [6, Chapter 7, Entry 22], and [37]. Note that

$\sum_{n}\frac{1}{n^{k}(n+1)^{k}}=\sum_{n<n_{2}}\frac{1}{n^{k}n_{2}^{k}}-\sum_{n<n_{2}}\frac{1}{n^{k}(n_{2}+1)^{k}}.$

If we write the values

$\sum_{n_{1}<\cdots<n_{r}}\frac{1}{(n_{1}+c_{1})^{k_{1}}\cdots(n_{r}+c_{r})^{k_{r}}},c_{1}\cdots,c_{r}\in\mathbb{N},$

as a sum of zeta values and a rational number, then we believe the
coefficients should have a poset interpretation.

  * •

Is there a generating function for the coefficients $d_{P,i}$ where $P$ runs
over all finite posets with $n$ points?

## Acknowledgements

We would like to express our gratitude to Antonio Arciniega-Nevarez and Marko
Berghoff for their valuable feedback on earlier versions of this document. We
also extend our thanks to Greta Panova and the anonymous reviewers for their
insightful comments. We are especially grateful to Hansol Hong for his
unwavering support and encouragement throughout the project. Additionally, we
would like to acknowledge the support provided by the faculty members of
Northwestern University and the CINVESTAV over the years. Their contributions
have been invaluable to the development of this work. We used the software
[27, 52] for several experiments.

##  7\. Funding

The first author’s research was made possible through the generous support of
the National Research Foundation of Korea (NRF) grant funded by the Korean
government (MSIT) (No. 2020R1C1C1A01008261). JLMC’s research was supported by
start-up funds from Michigan State University (MSU).

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