The algebraic-combinatorics tag has no wiki summary.

**6**

votes

**1**answer

177 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 ...

**0**

votes

**0**answers

34 views

### $B_k[1]$ 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 ...

**10**

votes

**1**answer

643 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 ...

**3**

votes

**0**answers

68 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 ...

**3**

votes

**0**answers

99 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 ...

**17**

votes

**5**answers

537 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 ...

**4**

votes

**0**answers

160 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?

**9**

votes

**1**answer

249 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} ...

**2**

votes

**0**answers

75 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) = ...

**6**

votes

**0**answers

300 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 ...

**2**

votes

**1**answer

190 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 ...

**1**

vote

**3**answers

196 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 ...

**2**

votes

**0**answers

55 views

### 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, ...

**11**

votes

**1**answer

239 views

### A Product Related to Unrestricted Partitions

Start with the product for unrestricted partitions:
(1+x+x$^2$+...)(1+x$^2$+x$^4$+...)(1+x$^3$+x$^6$+...)...
Now replace some of the plus signs with minus signs and expand the product into a ...

**2**

votes

**2**answers

237 views

### Computing the lexicographic indices of integer partition

If we order all the partitions of a integer in a lexicographic order, how can we compute the position of each partition in this order without having to explicitly list all other partitions that ...

**3**

votes

**2**answers

258 views

### Sandpile group corresponding to Abelian group

How we can prove each finite Abelian group is the sandpile group for some graph ?

**0**

votes

**1**answer

268 views

### Missing formula! [closed]

I am doing a project on group association schemes, in particular looking at the structure constant
$$p_{KL}^M = \#\{(x, y, xy) : x \in K, y\in L, xy \in M\}$$
where $K, L$ and $M$ are conjugacy ...

**13**

votes

**2**answers

794 views

### Has Reifegerste's Theorem on RSK and Knuth relations received a slick proof by now?

For the notations I am using, I refer to the Appendix at the end of this post.
Here is what, for the sake of this post, I consider to be Reifegerste's theorem:
Theorem 1. Let $n\in\mathbb N$ and ...

**8**

votes

**3**answers

459 views

### Arithmetic product of symmetric functions: why is it integral?

For every commutative ring $A$, let $\mathbf{Symm}_A$ be the ring of symmetric functions over $A$. Let $\mathbf{Symm}$ without a subscript denote $\mathbf{Symm}_{\mathbb{Z}}$.
We can define a ...

**15**

votes

**1**answer

519 views

### Conjectural identities for Young symmetrizers and Young-Jucys-Murphy elements

The following questions I have found in my own notes from about 3 years ago. Unfortunately, I lost much of the context; I believe I made these conjectures reading Okounkov-Vershik, arXiv:0503040v3, ...

**0**

votes

**2**answers

283 views

### Morse matching with 0-cells and (n-1)-cells

Suppose one has a Morse matching (acyclic matching) of a poset $P$ of rank $n$ in which the only unmatched cells are 0-cells and $(n-1)$-cells, the same number $k$ of each one.
If $P$ is connected ...

**10**

votes

**1**answer

243 views

### Reference request: Grothendieck groups of Hecke algebras at root of unity and symmetric functions

Let $\zeta$ be an $\ell^{\text{th}}$ root of unity, and consider $H_n(\zeta)$, the (finite) Hecke algebra of type A. One can consider a dual pair of Hopf algebras arising from this data, denoted ...

**8**

votes

**1**answer

262 views

### Plethysm of $\mathrm{QSym}$ into $\mathrm{QSym}$: can it be defined?

I will denote by $\Lambda$ the ring of symmetric functions, and by $\mathrm{QSym}$ the ring of quasisymmetric functions (both in infinitely many variables $x_1$, $x_2$, $x_3$, ..., both over $\mathbb ...

**9**

votes

**2**answers

267 views

### How exactly does Schützenberger promotion relate to Striker-Williams promotion?

SchĂ¼tzenberger promotion, studied (for example) in Richard Stanley, Promotion and Evacuation, 2009, is a permutation of the set of all linear extensions of a finite poset. Since one can identify the ...

**6**

votes

**2**answers

583 views

### What are the most important open problems in algebraic combinatorics? [closed]

I have seen the paper of Stanley http://math.mit.edu/~rstan/pubs/pubfiles/116.pdf
but it is quite old and many of the problems are solved. I would like to know the two or three biggest open problems ...

**3**

votes

**1**answer

185 views

### Semi-planar partition monoid/algebra

Here are some beginner questions on partition algebras...
I am trying to understand the monoid called $P_k$ in Tom Halverson, Arun Ram, Partition Algebras. For the sake of simplicity, let $k$ be a ...

**8**

votes

**3**answers

569 views

### bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n

Suppose $\lambda = (\lambda_1,\lambda_2,.....,\lambda_k)$ is a partition of $2n$ where $n \in \mathbb N$ satisfying the following conditions:
(1) $\lambda_{k} = 1$.
(2) $\lambda_{i} - \lambda_{i+1} ...

**2**

votes

**1**answer

444 views

### Combinatorial Inequality

Consider a set of $2^n-1$ non negative integers $S= ${$ a_{i,j}|1\le i\le n; 1\le j\le 2^{i-1} $} such that:
\begin{align}{}
1.\ \ &a_{i,j}\le 2^{n+1-i} \\\\
2.\ \ &a_{i,j}\le a_{i-1,j} \\\\
...

**1**

vote

**1**answer

180 views

### principal specialization of projective Schur functions

Is there a nice/known formula for the (some) principal specialization of a projective (P or Q) Schur polynomial? That is, I am talking about a projective analogue of Stanley's hook content formula ...

**4**

votes

**1**answer

173 views

### Is the “renormalized third comultiplication” on $\mathbf{Symm}$ integral?

Background:
For any commutative ring $R$, let $\mathbf{Symm}_R$ be the ring of symmetric functions in countably many variables $x_1$, $x_2$, $x_3$, ... over $R$. ("Symmetric functions" really means ...

**2**

votes

**2**answers

261 views

### Alternating sum of binomial coefficients times logarithm

Trying to find a closed form expression for the following sum, or an asymptotic expression in terms of well known functions (like the Gamma function, for instance).
Let $m,n$ be positive integers ...

**10**

votes

**1**answer

447 views

### Analogues of the Knuth and Forgotten equivalences on permutations: have they been studied?

Consider a totally ordered alphabet $A$ of $n$ letters. Let $W$ be the set of all words over $A$ which have no two letters equal. Then, for example, we can define the Knuth equivalence on $W$ as the ...

**20**

votes

**1**answer

727 views

### A strange sum over bipartite graphs

While mucking around with some generating functions related to enumeration of regular bipartite graphs, I stumbled across the following cutie. I wonder if anyone has seen it before, and/or if anyone ...

**2**

votes

**0**answers

322 views

### Software for Combinatorial Algebra sought

I am looking for software which helps me do straightforward tasks in combinatorial algebra. Let me give an example of what I mean by a straightforward task:
I have two graded (generally ...

**8**

votes

**2**answers

597 views

### Ubiquitous Zimin words

Let $w$ be a word in letters $x_1,...,x_n$. A value of $w$ is any word of the form $w(u_1,...,u_n)$ where $u_1,...,u_n$ are words. For example, $abaaba$ is a value of $x^2$. A word $u$ is called ...

**8**

votes

**1**answer

671 views

### Connected graded involutive bialgebras (Hopf algebras) sought (for Dynkin idempotent checking)

Again, there is a general and a concrete question:
Concrete question. Let $H$ be a connected graded bialgebra over a commutative ring $k$ with unity (where graded means $\mathbb N$-graded). Then, it ...

**3**

votes

**6**answers

247 views

### Do stunted exponential series give projections of a cocommutative bialgebra on its coradical filtration?

Let $k$ be a field of characteristic $0$.
Let $H$ be a cocommutative connected filtered bialgebra over $k$. ("Connected" means that the counit, restricted to the $0$-th part of the filtration, is an ...

**2**

votes

**1**answer

288 views

### Rational Binomial Identity

Can anyone give a reference, a proof, or a reference that explains why Maple can evaluate this identity mathematically correctly:
...

**6**

votes

**1**answer

236 views

### Simplices in convex polytopes

This question is a direct generalization of:
Counting the (additive) decompositions of a quadratic, symmetric, empty-diagonal and constant-line matrix into permutation matrices
Given a convex ...

**19**

votes

**2**answers

2k views

### Regular, Gorenstein and Cohen-Macaulay

All the statements below are considered over local rings, so by regular, I mean a regular local ring and so on;
It is well-known that every regular ring is Gorenstein and every Gorenstein ring is ...

**4**

votes

**1**answer

279 views

### Monotonicity of complete homogeneous symmetric polynomials

The complete homogeneous symmetric polynomials are defined as
$$
h_k (x_1, \dots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \cdots \leq i_k \leq n} x_{i_1} x_{i_2} \cdots x_{i_k}.
$$
For example,
$$
...

**5**

votes

**1**answer

289 views

### On MacMahon's conjecture and a Schur function identity

Recently I am reading Professor Bressoud's book "Proofs and confirmations"ă€‚And chapter 4 of his book is about using Schur functions to prove Macmahon's conjecture on symmetric plane partitions: ...

**12**

votes

**1**answer

1k views

### Sublattices of Young's Lattice

Young's Lattice is the lattice of inclusions of Young tableaux, or integer partitions.
In R. Stanley's Enumerative Combinatorics Vol. 1, Proposition 1.4.4., there is a generalization of integer ...

**2**

votes

**1**answer

207 views

### Reference for: the Bruhat-minimal permutations not less than a fixed permutation pi?

Let $\pi\in S_n$. I recently needed to understand the permutations $\rho$ such that $\rho\not\leq\pi$ in Bruhat order. Since there are $O(n!)$ of those I really wanted a description of the $O(n^2)$ ...

**6**

votes

**4**answers

3k views

### When is an algebra of commuting matrices (contained in one) generated by a single matrix?

Let C be an nxn matrix, then the polynomials in C (with appropriate coefficients) form an algebra of commuting matrices. I feel that I should know if the converse is true but I do not. So my first ...

**4**

votes

**0**answers

296 views

### Can the Littlewood-Richardson cone be used for combinatorial optimization?

The Littlewood-Richardson cone $LR_{n, k}$ consists of all $k-$tuples $(a_1, a_2, \dots, a_k)$ of real $n-$vectors with monotonically decreasing entries such that there exist $k$ $n \times ...

**6**

votes

**2**answers

668 views

### What is the most general “two in one row for A & in one column for B” theorem?

Let $A$ and $B$ be two Young tableaux, i. e. Young diagrams filled with the numbers $1$, $2$, ..., $n$ for some $n$ (not necessarily the same $n$). (They need not be semistandard.)
(a) (Etingof's ...