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146 views

### Evaluation of sum of factorials

Is there an evaluation of this sum (possibly involving gamma functions)? $k$ and $n$ are natural numbers and $x$ is real with $0<x<1$.
$$ \sum_{\substack{k=0\\n-k\text{ even}}}^n \frac{(-1)^{(n-...

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**0**answers

244 views

### Coefficients in expansion of a classical symmetric polynomial

If we expand \begin{equation} P_3(x_1,\ldots, x_n):=\Pi_{1\leq i<j<k\leq n} (x_i+x_j+x_k), \end{equation} then \begin{equation} P_3 = \sum_\alpha \sum_{\mathcal{O}(\alpha)} c_\alpha x_1^{\...

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**2**answers

100 views

### Number of Skew Symmetric Matrices of fixed rank

The number of symmetric matrices of order $n$ and rank $r$ over finite fields has been counted e.g.
http://www.math.clemson.edu/~kevja/REU/2004/SymmetricRankRMatrices.pdf
Is the number of skew-...

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**1**answer

115 views

### Cross between the nil-Hecke ring and the group ring of a Coxeter group

A Coxeter system $(W,S)$ has a set of generators $S=\{s_1,s_2,\ldots\}$ and the Coxeter group $W$ is determined by relations of the form $(s_is_j)^{m_{ij}}=1$ for some integers $m_{ij}$, where $m_{ii}=...

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**3**answers

286 views

### Combinatorial aspects of continued fractions

Recently, I got interested in the study of the combinatorial aspects of continued fractions. Precisely, I read of the following lemma of Flajolet (see here):
Lemma. It holds
$$\sum_{\omega} \nu(\...

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**1**answer

212 views

### On factorization theorem of toric birational morphisms

Let $X_{Σ′}\to X_{Σ}$ be a toric birational morphism between smooth and complete toric varieties induced by a regular subdivision $Σ′\leq Σ$, i.e. every cone in $Σ′$ is contained in a cone in $Σ$ and ...

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54 views

### Rank-unimodality and Sperner property of higher dimensional partitions

I did a google-search but have not been able to find much reference on this problem. So I am asking it here hoping to get some information.
Consider the set of all 4-dimensional Ferrer's diagram ...

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17 views

### How to find a subset of a matrix that has minimum condition number? [duplicate]

Suppose matrix $A$ is consist of M column vectors, how can we find a subset $B$, consisting of N column of $A$ (N＜M), that has minimum condition number (the ratio of maximum singular value by minimum ...

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**14**answers

1k views

### Applications of Representation Theory in Combinatorics

What are the examples of interesting combinatorial identities (e.g. bijection between two sets of combinatorial objects) that can be proved using representation theory, or has some representation ...

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**3**answers

500 views

### The number of submodules of $\mathbb{Z}_q^n$

Observe $\mathbb{Z}_q^n = \mathbb{Z}_q \times \cdots \times\mathbb{Z}_q$ as a module over $\mathbb{Z}_q\equiv\mathbb{Z}/q\mathbb{Z}$, for general $q$.
I am interested in the following questions:
How ...

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**1**answer

271 views

### A remarkable sum over partitions

While studying some seemingly unrelated topological questions, I have experimentally discovered what appears (to me) to be a remarkable sum over partitions. I was wondering if anyone knows how to ...

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**1**answer

103 views

### When is the diagonal of a rational bivariate power series again rational

Given a rational bivariate power series $F(x,y)=\sum{a_{n,m}x^ny^m}$, the diagonal function $G(t):=\sum{a_{n,n}t^n}$ is known to be algebraic, although not rational in general. I was wondering if ...

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**1**answer

96 views

### Is there a nice form for the Frobenius characteristic of a border shape character?

Let $\chi^V$ be the character of a border strip Specht module, i.e. a Specht
module for a skew tableau that contains no $2 \times 2$ square. I know that
the Frobenius characteristic of $\chi^V$ is ...

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**2**answers

373 views

### Sums of reciprocals of products of factorials

Let $d,m, r$ be positive integers, and define
$$
S = \left\{ (i_1, i_2, \dots, i_m) \in {\bf Z}_{+}^{m} \left | \sum_j i_j = d; \& \forall j, i_j \leq r \right. \right\};
$$
Here ${\bf Z}_+$ ...

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85 views

### Abelian centralizer groups (CA-groups)

I am searching for all information about CA-groups [abelian centralizer groups] and i just found a German book [Huppert] and Nilpotent Centralizer group of Suzuki in 44 pages and Group theory book of ...

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113 views

### Counting square zero forms over finite fields

Let $p$ be an odd prime and let $R=\Lambda_{\mathbb{F}_p}[x_1,\dots,x_n]$ be the exterior algebra on $n$ generators over the finite field with $p$ elements. This is a graded-commutative ring.
Is ...

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**1**answer

160 views

### Littlewood-Richardson-Type Rule for Restriction from $S_{2n}$ to $S_{2(n-t)} \times (S_2 \wr S_t)$

It is well-known that the Littlewood-Richardson coefficient $c^{\nu}_{\lambda \mu}$ is the number of times the irreducible representation $V_\lambda \bigotimes V_\mu$ of the product of symmetric ...

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**2**answers

300 views

### Free $k[x_1, \dots, x_n]^{S_n}$-module?

Let $\text{char}\,k = 0$ and $n \ge 2$. What is the easiest way to see that $k[x_1, \dots, x_n]$ is a free $k[x_1, \dots, x_n]^{S_n}$-module with basis$$x_2^{m_2}x_3^{m_3} \dots x_{n-1}^{m_{n-1}} x_n^{...

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89 views

### Properties of Coefficients of Order Polynomials [closed]

I am working on a problem involving determining the order polynomial $\Omega_P(k)$ of a partial order $P$, which counts the number of order-preserving transformations/maps from $P$ to the $k$-chain (...

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**2**answers

230 views

### Dickson/determinant type polynomial (updated)

For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation}
...

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**1**answer

110 views

### bijection between S-modules and Schur functors

Given a $\mathbb{S}$-module, we can construct a Schur functor, which is an endo functor in the category of vectorspaces. But given a Schur functor, how can we get back the $\mathbb{S}$-module? In ...

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376 views

### Product of a Schubert polynomial and a double Schubert polynomial

Let $S_u(x)$ be a Schubert polynomial and let $S_v(x;y)$ be a double Schubert polynomial. Then their product can be expressed in terms of the double Schubert polynomials as
$$S_u(x)S_v(x;y)=\sum_w{c_{...

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195 views

### An (open?) problem about a sequence of nested principal sub-matrices and their determinants

Problem: Let $A$ be a $n \times n$ integer matrix, $\det(A) = \pm 1$. Under which conditions there exist a nested sequence of principal submatrices of size $n$ such that they all have determinant $\pm ...

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**1**answer

154 views

### Generating function for $t$-residues of partitions using Heisenberg + $\hat{sl_t}$ representation theory

Recall that for $t\geq2$, a partition is a $t$-core if none of its hooklengths is divisible by $t$. It is known that the $t$-cores are parametrized by ${\mathbb Z}^{t-1}$. More precisely, let $(n_0,\...

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126 views

### A vector version of the Segre embedding: what is the kernel of the ring map?

TL;DR version.
Given a commutative ring $\mathbf{k}$ and $n+m$ "generic" vectors $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n, \mathbf{y}_1, \mathbf{y}_2, \ldots, \mathbf{y}_m$ in $\mathbf{k}^k$ ...

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456 views

### Nekrasov-Okounkov hook length formula

I am now reading the paper An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths by Guo-Niu Han. The author rediscovered what he calls Nekrasov-Okounkov ...

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269 views

### Deforming a basis of a polynomial ring

The ring $Symm$ of symmetric functions in infinitely many variables is well-known to be a polynomial ring in the elementary symmetric functions, and has a $\mathbb Z$-basis of Schur functions $\{S_\...

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101 views

### Chern character of Schubert structure sheaf

Let $X_\lambda \subset Gr = Gr(k,n)$ be a Schubert variety in the Grassmannian and $\mathrm{ch} : K_0(Gr) \otimes \mathbb{Q} \to A^\bullet(Gr) \otimes \mathbb{Q}$ the Chern character isomorphism.
Is ...

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**1**answer

202 views

### Structure of the stabilizer of a vertex-neighborhood of a vertex-transitive graph

Given a simple, undirected graph and a vertex $v$ of the graph, let $L_v$ denote the set of automorphisms of the graph that fixes the vertex $v$ and each of its neighbors. When the graph is vertex-...

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157 views

### Transitivity for Schutzenberger involutions on standard Young tableaux

Let $\lambda$ be a partition of $ n$. Let $ SYT(\lambda) $ denote the set of standard Young tableaux of shape $ \lambda $.
For $ i = 1, \dots, n $, let me define permutations $ S_i $ of the set $ ...

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83 views

### combinatorial ergodicity and promotion

According to J. Propp, T. Roby, and (I believe) others, a cyclic action on a finite set $S$ given by a bijection $\zeta: S \longrightarrow S$ is said to be ${\it ergodic}$ with respect to a statistic ...

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1k views

### “Nyldon words”: understanding a class of words factorizing the free monoid increasingly

BACKGROUND.
Let me first introduce some classical definitions, which appear, e.g., in §5 of Lothaire's Combinatorics on Words, in §5.1 of Reutenauer's Free Lie algebras, and in §6.1 of Victor Reiner'...

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164 views

### Anti-arithmetic product of symmetric functions: (why) is it integral?

This is an analogue of MathOverflow question #138148. Indeed it is so analogous that I wrote the following by copypasting said question and making the necessary changes.
For every commutative ring $A$...

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88 views

### Dimension of the sum of images of transpose

$\newcommand{\rank}{\operatorname{rank}}\newcommand{\im}{\operatorname{im}}$
Given $A,B\in M_{n\times n}(k)$, define $\rank(A,B):=\dim(\im A+\im B)$. I'm looking for results regarding relationships ...

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266 views

### An angle-doubling trick of Kirillov and Berenstein [closed]

Kirillov and Berenstein, in their article "Groups generated by involutions, Gelfand-Tsetlin patterns and combinatorics of Young tableaux" (available at math.uoregon.edu/~arkadiy/bk1.pdf), present a ...

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387 views

### Is there a hyperplane avoiding two independent sets?

Let $V$ be a vector space over a field with $5$ elements, $A,B \subseteq V$ independent subsets. Must there be a subspace of $V$ of codimension 1 disjoint from $A \cup B$?

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353 views

### An identity for elementary symmetric functions

Trying to understand a result in a representation theoretical paper, I realized that it implies the following elementary identity for symmetric functions. My question is whether this identity is true, ...

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**1**answer

253 views

### Über theorem on unavoidable patterns?

Let $A$ be an alphabet of $k$ symbols,
and $p$ a pattern.
An example of a pattern is $p=XX$, where $X$ is any finite
string of symbols from $A^+$.
Avoiding $p$ is avoiding any subword repeated twice ...

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### $B_k[1]$ sets with smallest possible $m = max B_k[1]$ for given $k$ and $n = |B_k[1]|$ elements

Sidon sets are sets $A \subset \mathbb{N}$ such that for all $a_j,b_j \in A$ holds
$$a_1+a_2=b_1+b_2 \iff \{a_1,a_2\}=\{b_1,b_2\}$$
Thus if you know the sum of two elements, you know which elements ...

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**1**answer

812 views

### Two to the power of a triangular number: bijections

The numbers $2^{n(n+1)/2}$ come up in various enumerative contexts. In addition to the trivial example (bit-strings of length $n(n+1)/2$) and the old example of domino tilings of Aztec diamonds (...

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128 views

### Counting perfect matchings with integrals

Has anyone used the Joni-Rota-Godsil integral formula (see details below) to count perfect matchings of square-grid graphs, Aztec diamond graphs, hexagon-honeycomb graphs, etc.? (Or even just to ...

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229 views

### Triangulations of special polyhedra

Let $A_1,A_2,A_3 \in \mathbb{N}^3$ be three points in space all lying in some plane $x+y+z=d$ where $d$ is a positive integer. If $\{e_1,e_2,e_3\}$ is the standard basis in $\mathbb{R}^3$, we can ...

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878 views

### Why is the right permutohedron order (aka weak order) on $S_n$ a lattice?

This is one of those things I never expected to be hard until I tried to prove it. Why is the right permutohedron order (a.k.a. weak Bruhat order, a.k.a. weak order -- not to be confused with the ...

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**0**answers

244 views

### Littlewood-Richardson rule for Schubert polynomials

What is the current state of the problem of finding a combinatorial rule for multiplying two Schubert polynomials? Is the problem still open?

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**1**answer

607 views

### Why are the power symmetric functions sums of hook Schur functions only?

One interesting fact in symmetric function theory is that the power symmetric function $p_n$ can be written as an alternating sum of hook Schur functions $s_{\lambda}$:
$$
p_n = \sum_{k+\ell = n} (-1)^...

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87 views

### Lagrangean equations for the generating function of quadrangulations

Let $M(z)$ be the generating function of edge-rooted connected quadrangulations, with $z$ marking the number of edges. I derived the following Lagrangean equations for $M(z)$:
$$M(z) = \psi(L(z)),~\...

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419 views

### On the number of polynomials that divide $x^{q-1}-1$ in some subspaces of $\mathbb F_q[x]$

Let $\mathbb F_q$ be a finite field (where $q$ is in general a power of a prime), and let $e, k$ be positive integers with $k \leq e < q-1$. Let $f_0(x), \ldots,
f_k(x) \in \mathbb F_q[x]$ be ...

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**1**answer

274 views

### Question about a proof in Graham and Lehrer's “Cellular algebras”

I'm sorry if this question is too basic for MO. I'm reading a paper by Graham and Lehrer "Cellular algebras" and have trouble understanding one step in a proof of a crucial theorem. I suppose that the ...

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**3**answers

234 views

### Constructions of $2-(v,3,3)$-designs

I am looking for ways to construct an infinite family of designs with parameters $2-(v,3,3)$ and apart from some doubling-type recursive constructions (such as in this paper) I haven't found anything ...

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### Rank generating functions on a graded poset and a linear extension of it

Let $A$ be a possibly infinite ensemble and let $\leq$ be a partial order between elements of $A$. We denote $P_{\leq}=(A,\leq)$ the corresponding poset. Furthermore, we suppose that $P$ is graded, i....