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# Questions tagged [algebraic-combinatorics]

For problems involving algebraic methods in combinatorics (especially group theory and representation theory) as well as combinatorial methods in abstract algebra.

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### Can you help me finding another non trivial families of groups indexed by an infinite countable ordered set? [closed]

Symmetric groups are indexed by nonnegative integers and Young subgroups are indexed by composition of integer. Some Coxeter groups are indexed by integer. Can someone help me finding another non-...
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### Proving Pascal's Identity for more than two terms [duplicate]

This involves a slightly different version of Pascal's Identity but is still based on it. Prove that $${a \choose k} + {a+1 \choose k} + \dots {a+n \choose k} = {a+n+1 \choose k+1}$$ I have tried to ...
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I have seen a lot of papers discussing whether various permutation classes are Wilf equivalent to each other. I wonder if we could solve such problems in general with computers. More rigorously, let $\... 1 vote 0 answers 16 views ### Number of semi-standard Young tableaux of shape$\lambda$with some entries fixed Given a partition$\lambda$, the number of semi-standard Young tableaux (SSYT) of shape$\lambda$with maximum entry$n$is given by \begin{equation} \prod_{1\leq i<j\leq n} \frac{\lambda_i-\... 2 votes 0 answers 15 views ### Derivation of the Macdonald operator$D_{n}(X;q,t)$Since I first encountered Equation (3.2) on p.315 of Macdonald's Symmetric functions and Hall polynomials, I have wanted to know where it comes from. So how does one derive the operator \begin{... -3 votes 1 answer 124 views ### Proving combinatoric identities with the inclusion and exclusion principle Let V be a finite set and let the number$a(F) \in\Bbb R$for$ F \subseteq V$. We define the numbers $$b(G, E) = \sum_{E\subseteq F\subseteq G} (-1)^{|G-F|} a(F)$$ where$E\subseteq G\subseteq V$... 0 votes 1 answer 32 views ### Clique-coclique bound in association scheme Let$A_0=I,A_1,\dots,A_k$be the assocation matrices of a$k-$class association scheme$R_0,\dots,R_k$on a set$X$. Let$K \subset \{0,1,\dots,k\}$. We say a subset$Y \subset X$is a$K-$coclique ... 2 votes 0 answers 139 views ### Prove$\sum_{n \ge 0} (n+1)^{n-1}\frac{x^n}{n!} =(1- \sum_{n \ge 1} (n-1)^{n-1} \frac{x^n}{n!}) ^{-1}$Is it possible to prove that $$\sum_{n \ge 0} (n+1)^{n-1}\frac{x^n}{n!} =(1- \sum_{n \ge 1} (n-1)^{n-1} \frac{x^n}{n!}) ^{-1}$$ Where$(n-1)^{n-1} := 1$for$n=1$Idea:use the Lagrange inversion ... -1 votes 1 answer 77 views ### How many ways can someone choose a permutation$w $and color each one of the integers [n] so that the minimum element of every cycle of w is white? In how many ways can someone choose a permutation$w \in S_n$and color each one of the integers$1,2,\ldots,n$white, yellow or blue so that the minimum element of every cycle of$w$is white? ... 1 vote 1 answer 157 views ### the number of sequences is equal to the number of permutations Consider the product$A_n = \left\{1\right\} \times \left\{1,2\right\} \times \cdots \times \left\{1,2,\ldots,n\right\}$. For$\sigma = (a_1, a_2, . . . , a_n) \in A_n$, define the set of descents$\... 55 views

### Forgotten Symmetric Functions

Let A be a partition of n of length $\lambda$. Define the forgotten symmetric function $f_\lambda$ by $f_\lambda = \varepsilon_\lambda \omega(m_\lambda)$, where $\varepsilon_\lambda = (-1)^{n-l}$ as ...
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### Prove that the coefficients of F(x) are positive integers [closed]

Consider the formal power series $$F(x) = \sum_{k\ge0}(x+x^2-x^3)^k$$ How can I show that the coefficients of F(x) are positive integers? I wrote the F as $F(x) = 1/(1-x-x^2 +x^3) = 1/((x-1)^2(x+1))$ ... 37 views

### Can we characterize the “associate classes” of a unipotent quasi-commutative quasigroup as some combinatorial design?

$I_n$ is the $n \times n$ or order $n$ identity matrix, $J_n$ is the order $n$ matrix of all ones, and $n \in \mathbb{Z}^+$. We define a Latin square $\mathcal{L_n}$ to be a set of $n$ permutation ...
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### Are there interesting combinatorial proofs which use more sophisticated grouping than sign-reversing involutions?

There are many combinatorial proofs which establish interesting identities by designing suitable "sign-reversing involutions" on a set of relevant signed objects. For example, Benjamin and ...
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### Evaluating character functions

Immanants generalize the notion of determinant and permanent, and it is defined as $$Imm_{\lambda}(A)=\sum_{\sigma \in S_n}\chi_{\lambda}(\sigma)\prod_{i=1}^{n}a_{i,\sigma(i)}$$ where $\chi_{\lambda}$ ...
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### What is the Lie theoretic interpretation of conjugate of a partition?

For a partition $\lambda$ it is very well-known operation to take its conjugate partition $\lambda'$ which is obtained by transposing the Young diagram of $\lambda$. A partition $\lambda$ can be ...
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### generating function of finite sum involving $\left(\!\!\binom{n}{k}\!\!\right)$ [closed]

Notation: $$\left(\!\! \binom{n}{k}\!\!\right)={n+k-1 \choose k}=\frac{(n+k-1)!}{k!(n-1)!}$$ where $n!$ is the factorial, i.e. $1\cdot 2\cdots n.$ Let $n,N\in \mathbb{Z}_{>0}.$ I'm stuck at ...
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### Binomial coefficient identities from h-vector of a simplicity complex. [duplicate]

My question is to show the below equality $$\sum_{j=0}^{d}(-1)^{j}\binom{d}{j}\binom{d-j}{i}=0$$ when $d>i \geq 0$ for any integers $d,i$. This inequality is came from Stanley's note. Given an $f$-...
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### Operations with Circulant Matrix using GAP

I am newbie using GAP software. I need to know how to use GAP software for algebraic computations with circulant matrix. Some examples would suffice. Just for clarity Circulant Matrix: In linear ...
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### The Schwartz genus of a fibration

I'm working on the topological complexity of a path connected space, which is a particular case of the sectional category (or the Schwartz genus), but all the articles refer to an other article named &...
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### change of summation, Hermite polynomial formula from generating functions

Now given the exponential generating function of the Hermite polynomial $$e^{ux-(1/2)u^2} = 1+\sum_{n=1}^\infty \dfrac{u^n}{n!}h_n(x),$$how do I derive the closed form formula of the Hermite ...
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### On the dimension of an abstract simplicial complex built from minimal vertex covers of a finite simple graph

Let $G$ be a finite simple graph on a vertex set $\{x_1,...,x_n\}.$ Let us call a vertex cover of $G$ to be minimal if none of its proper subset is a vertex cover. Let $C_1,...,C_h$ be the collection ...