# Questions tagged [algebraic-combinatorics]

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### Canon in algebraic combinatorics and how to study

1) In subjects such as algebraic geometry, algebraic topology there is a very basic standard canonical syllabus of things one learns in order to get to reading research papers. Is there a similar ...
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### Algebraic ode of exponential generating series

Let $G(z)$ be a rational function. So if we have a series $$S(x):=\sum_{n}a_n x^n$$ where $$a_n = \prod_{i=1}^{n}G((i-1)h)$$ We can conclude that the series satisfies a Linear differential ...
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### A special class of weighted Motzkin paths

Consider Motzkin paths with the following weight: All up-steps and the horizontal steps on height $0$ have weight $1$, all down-steps have weight $t$ and the horizontal steps on even heights have ...
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### $2$-adic valuation of Schur $P$-functions in the power-sum basis

For a partition $\lambda$, let $P_\lambda$ be the Schur $P$-functions (case $t=-1$ of Hall-Littlewood symmetric functions) and let $p_\lambda=p_{\lambda_1}p_{\lambda_1}\cdots p_{\lambda_k}$ be the ...
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### Geometric or combinatorial interpretations of the (weak) Bruhat order?

$\DeclareMathOperator\Inv{Inv}$The weak Bruhat order on the symmetric group has a straightforward combinatorial interpretation: Consider a set of labelled balls $1,2,\dotsc,n$. Then for two ...
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### $B_k$ sets with smallest possible $m = \max B_k$ for given $k$ and $n = \lvert B_k\rvert$ 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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### What is a toric lattice? [closed]

What is a toric lattice? and how can I construct one in Macaulay2 and compute its basis? is there any alternative method to make one? Since I went through the whole ...
Let $S_n$ be the symmetric group. Consider $$D:=\sum_{\sigma\in S_n} \text{sgn}(\sigma)\cdot \det\begin{pmatrix}1 & a_{\sigma(1)}-0 & (a_{\sigma(1)}-0)^2 & \cdots & (a_{\sigma(1)}-0)^{... 3answers 359 views ### Given a positive integer n, some straight lines and lattice points such... Prove that the number of the lines is at least n(n+3) I was trying to get an answer on MathSE long ago and now I got it. Given a positive integer n and some straight lines in the plane such that none of the lines passes through (0,0), and such that ... 0answers 74 views ### Littelmann Path model and RSK e and f operators The Littelmann path model defines e_i and f_i operators which correspond in type A to the e_i and f_i operators on semistandard Young tableaux (i.e. since they both are different ways of ... 0answers 194 views ### Can Matsumoto's theorem for the symmetric group be proved using a monovariant? This is a question that can be asked for any Coxeter group, but for the sake of simplicity I will restrict myself to symmetric groups. Recall the main definitions: Let n be a nonnegative integer. ... 0answers 98 views ### How to classify rings by combinatorial structures? There are many ways to encode information about algebraic structures such as groups, rings, etc... in combinatorial form. For example the Cayley graph of a group with a subset of generators, or the ... 2answers 416 views ### Chip-firing clocks Let G be some outdegree-regular directed graph with n vertices and let H be the Laplacian of G, so that the rows of H correspond to chip-firing moves. I’m interested in linear functions f ... 0answers 46 views ### Making the entries of a matrix positive I am considering two slightly more relaxed version of the question asked here: https://math.stackexchange.com/questions/119034/making-the-entries-of-a-matrix-positive The two questions are: Question 1:... 1answer 205 views ### proof of result from Ian Macdonald's paper "A New Class of Symmetric Functions" I'm currently working my way through Ian MacDonald's somewhat seminal 1988 paper entitled "A New Class of Symmetric Functions" in Seminaire Lotharingien B20a, pp. 131–171 (EuDML). I'm fine ... 0answers 142 views ### Does every finite lattice embed into a finite Eulerian lattice? A finite Boolean lattice is a lattice isomorphic to the subset lattice of a finite set. Every Boolean lattice is Eulerian, namely, a graded lattice L such that \mu(a,b) = (-1)^{|b|-|a|} for all a,... 0answers 340 views ### Two majs for standard Young tableaux? Let \lambda be a partition of n, and consider its set of standard Young tableaux (SYTs): bijective fillings of the Young diagram of \lambda, written in English notation, with the numbers 1 ... 0answers 66 views ### Form on symmetric functions and their q,t- analogues [Notations are as in Macdonald's Symmetric Functions and Hall Polynomials] The space of symmetric functions \Lambda_{\mathbb{Q}} has a bilinear form defined by  (p_\lambda, p_\mu)= z_\lambda \... 2answers 442 views ### Characterization of the family of simple groups PSL(2,q) by tensor multiplicity Let G be a finite group and (\chi_i) its irreducible characters. Then \forall i,j,k, \exists!n_{i,j}^k \in \mathbb{N}_{\ge 0} such that$$\chi_i \chi_j = \sum_k n_{i,j}^k \chi_k.Let the ... 0answers 58 views ### Double Schur function expansion In literature, I have seen the weighted Hurwitz number N_{g,n}(d_1 , d_2 \ldots , d_n) which are symmetric number and they can be written as double Schur function expansion. \begin{align} \label{eq:... 0answers 165 views ### Relationship between crystal root operators and usual e_i, f_i? Suppose I am working in a symmetrizable Kac–Moody Lie algebra \mathfrak{g}. Let e_1,\dotsc,e_n,f_1,\dotsc,f_n denote the usual Chevalley generators of \mathfrak{g}. Let V be a highest weight ... 0answers 171 views ### Word/Loop in L(\Lambda) Let \mathfrak{g} be a symmetrizable Kac-Moody algebra, with Chevalley generators e_i,f_i (i=1,...,n). Let L(\Lambda) denote the irreducible module with highest weight \Lambda. Let v denote ... 1answer 141 views ### Generalised operad structures We can naively consider an operad as a collection \{P(n)\}_{n\geq 0} of vector spaces P(n) consisting of "functions" with n inputs and one output, equipped with a number of ... 1answer 1k 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 ... 1answer 272 views ### Another characterization of matroids Has anyone seen the following characterization of matroids? Let \Delta be a simplicial complex on finite ground set E. Then \Delta is a matroid complex if and only if, for every X\subseteq E ... 1answer 636 views ### Number of conjugacy classes of finite reductive groups Let G be a connected reductive group over \mathbb{Z}. Let c_{G(\mathbb{F}_q)} be the number of conjugacy classes of G(\mathbb{F}_q). Question: Is it true that c_{G(\mathbb{F}_q)} is a quasi-... 1answer 166 views ### Is this Laurent phenomenon explained by invariance/periodicity? In Chapter 4 of his Tracking the Automatic Ant, David Gale discusses three families of recursively defined sequences of numbers, all due to Dana Scott and inspired by the Somos sequences: Sequence 1. ... 0answers 164 views ### Is there are good algebraic model of random n-hypergraphs? Suppose F is a finite field and -1 is a square in F. Let E be the binary relation on F where (a,b) \in E iff a - b is a square. Then (F,E) is called a Paley graph. Paley graphs are ... 0answers 104 views ### Polynomiality of the inverse of equally weighted Varchenko matrices attached to hyperplane arrangements Let d\in \mathbb{Z}_{\ge 1}, let \sigma = (H_i)_{i\in \mathcal{I}} be a finite hyperplane arrangement in \mathbb{R}^d, where H_i\subset \mathbb{R}^d is a hyperplane for i\in \mathcal{I} (the ... 0answers 100 views ### Can equality of chromatic symmetric functions of two trees be checked in polynomial time? Stanley defined chromatic symmetric functions (CSF) in 1995 (Advances in Math) where he conjectured that trees can be distinguished by their CSF. However, tree isomorphism is decidable in P (... 2answers 750 views ### Do you know an elegant proof for this expression for a Schur function? I know that the identity s_\mu = \sum_{\mu-\lambda \text{ is a horizontal strip}} \;\sum_{\alpha\vdash|\lambda|} \frac{\chi^\lambda_\alpha}{z_\alpha} \prod_i(p_i-1)^{a_i} $$holds. Here \alpha=1^{... 2answers 233 views ### Is there a noncommutative simple fusion ring? A fusion ring \mathcal{F} is given by a finite set B = \{b_1,b_2, \dots, b_r \} such that b_i b_j = \sum_k n_{i,j}^k b_k with n_{i,j}^k \in \mathbb{Z}_{\ge 0}, satisfying axioms slightly ... 2answers 242 views ### Combinatorial representation of function Let f(x, y, z) is the number of distinct ways of representing x as a sum of at most y positive integers that are all smaller or equal to z. Moreover, If yz < x, then the function gives 0.... 1answer 341 views ### Generalization of symmetric functions A n-variable function f is a symmetric function if$$f(x_1,x_2, \ldots, x_n) = f(x_{\sigma(1)}, x_{\sigma(2)}, \ldots, x_{\sigma(n)})$$for every permutation \sigma \in S_n. In particular, if f... 1answer 2k views ### Updates to Stanley's 1999 survey of positivity problems in algebraic combinatorics? [I am a co-moderator of the recently started Open Problems in Algebraic Combinatorics blog and as a result starting doing some searching for existing surveys of open problems in algebraic ... 0answers 105 views ### A combinatorial proof of an identity of partitions (Macdonald I.5) This is a statement from Symmetric Functions and Hall Polynomials by Macdonald: \sum_{x\in \lambda} (h(x)^2-c(x)^2)=|\lambda|^2 where \lambda denotes a partition or a Young diagram, and for each ... 4answers 4k views ### How to constructively/combinatorially prove Schur-Weyl duality? How is Schur-Weyl duality (specifically, the fact that the actions of the group ring \mathbb{K}\left[ S_{n}\right]  and the monoid ring \mathbb{K}\left[ \left( \operatorname*{End}V,\cdot\... 1answer 841 views ### Open problems concerning all the finite groups What are the open problems concerning all the finite groups? The references will be appreciated. Here are two examples: Aschbacher-Guralnick conjecture (AG1984 p.447): the number of conjugacy ... 4answers 5k 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 ... 0answers 69 views ### Bijections between binary sequences and primitive elements in a finite field [duplicate] Let n>1 be a natural number. We call a binary sequence (b_1,\ldots, b_n)\in \{0,1\}^n rigid if it is not a proper power of a sequence of shorter length. So for example (0,1,0,1) = (0,1)^2 ... 2answers 1k views ### Inequality for hook numbers in Young diagrams Consider a Young diagram \lambda = (\lambda_1,\ldots,\lambda_\ell). For a square (i,j) \in \lambda, define hook numbers h_{ij} = \lambda_i + \lambda_j' -i - j +1 and complementary hook numbers ... 1answer 134 views ### Digraphs with unique walk of length k between any two vertices Let G be a digraph such that there is an unique directed walk of length k between any two vertices. Equivalently, if A is the adjacency matrix of G, then A^k is the matrix with all entries ... 0answers 72 views ### Subalgebras of a polynomial ring carved out by (families of) coefficient equalities Let \mathbf{k} be a field, and let P=\mathbf{k}\left[ x_{1},x_{2} ,\ldots,x_{n}\right]  be a polynomial ring over \mathbf{k} in n variables x_{1},x_{2},\ldots,x_{n}. Alternatively, P can ... 2answers 391 views ### Abundancy index and non-solvable finite groups Let \sigma be the sum-of-divisors function. A number n is called abundant if \sigma(n)>2n. Note that the natural density of the abundant numbers is about 25 \%. The abundancy index of n ... 2answers 417 views ### On the sum of the subgroup orders of a finite group Let G be a finite group. Consider the function providing the sum of the subgroups orders$$\sigma(G) = \sum_{H \le G} |H|. Note that if $C_n$ is cyclic of order $n$ then $\sigma(C_n) = \sigma(n)$, ...
Start with the product for partitions into parts congruent to $1$ or $4$ modulo $5$: $(1 + x + x^2 + x^3 + ...)(1 + x^4 + x^8 + x^{12} +...)(1 + x^6 + x^{12} + x^{18} +...)$... Now replace some of the ...