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        Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

        2
        votes
        0answers
        17 views

        About relation between Kostka numbers and Littlewood-Richardson coefficient

        The fact that Kostka numbers equals to Littlewood-Richardson coefficients for some partitions is already known $\colon$ \begin{align} K_{\lambda \mu} = c_{\sigma \lambda}^\tau \end{align} where $\...
        6
        votes
        0answers
        61 views

        Numbers where there is a unique group with integral character table

        Call a number $n$ special in case there is a unique group of order n whose character table consists of only integers. This should be equivalent to the condition that the number of conjugacy classes ...
        25
        votes
        1answer
        492 views

        Number of irreducible representations of a finite group over a field of characteristic 0

        Let $G$ be a finite group and $K$ a field with $\mathbb{Q} \subseteq K \subseteq \mathbb{C}$. For $K=\mathbb{C}$ the number of irreducible representations of $KG$ is equal to the number of conjugacy ...
        11
        votes
        3answers
        430 views

        Which partitions realise group algebras of finite groups?

        Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $\mathbb{C}$). Given a partition $p=[a_1,...,a_m]$ of an integer $n$. We can identify $p$ with the ...
        3
        votes
        0answers
        22 views

        Maximal number of sets, obtained as intersections of semiintervals of $k$ linear orders

        Given a finite set $S$ with $n$ elements, and a fixed small $k$ (say $k=3$), how to find $k$ linear orders $\leq_1, \dots, \leq_k$ on $S$, such that the number of feasible subsets of $S$ is ...
        6
        votes
        0answers
        54 views

        How many maximal length Bruhat paths from $u$ to $w$ can there be?

        I've been doing some work with saturated Bruhat paths in a Coxeter group between two elements $u\leq w$. It seems to me that if $\ell(u) =0$, then there are at most $\ell(w)! $. I haven't tried to ...
        0
        votes
        1answer
        49 views

        Expected sum of chosen coordinates in a random subset of a Hamming hypercube

        Let $S$ = $\{v_1, v_2, ..., v_n\}$ denote a random subset of a Hamming hypercube of dimension $d$, where $n = |S|$ and $n \leq 2^d$. If $v_i$ = $\langle x^i_1x^i_2... x^i_d\rangle$ for all $i \in [1,n]...
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        votes
        0answers
        51 views

        Classical combinatorics? [on hold]

        For a problem in quantum mechanics I need to solve some combinatorics problem. It seems to me to be a natural question so people who know the field will be able to answer I think. How many ways are ...
        4
        votes
        1answer
        121 views

        Nonlinear boolean functions

        Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements. I wonder if there is any known algorithm/construction that, given any $n\geq 1$, returns a boolean function $f:\mathbb{F}^n_2\rightarrow \...
        1
        vote
        1answer
        173 views

        A generalization of Landau's function

        For a given $n > 0$ Landau's function is defined as $$g(n) := \max\{ \operatorname{lcm}(n_1, \ldots, n_k) \mid n = n_1 + \ldots + n_k \mbox{ for some $k$}\},$$ the least common multiple of all ...
        2
        votes
        1answer
        69 views

        Why Triangle of Mahonian numbers T(n,k) forms the rank of the vector space?

        I am looking for an explanation of why Triangle of Mahonian numbers T(n,k) form the rank of the vector space $H^k(GL_n/B)$? With respect to the property of Kendall-Mann numbers where the statement ...
        9
        votes
        1answer
        139 views

        Reference Request: Length of a reflection in a Coxeter group can be achieved by symmetric word

        In a given coxeter group $(W,S)$, a reflection is an element of $W$ that can be written with a symmetric word in the generators $S$. In multiple sources, I found the following formula: $$ \mathrm{...
        10
        votes
        2answers
        252 views

        Almost graceful tree conjecture

        The graceful tree conjecture is the following statement: for any tree $T = (V, E)$ with $|V| = n$ there is a bijective map $f: V \to [n]$ such that $D = \{|f(x) - f(y)| \mid xy \in E\} = [n - 1]$. ...
        10
        votes
        2answers
        358 views

        Find the tight upper bound of $\sum_{i=1}^n \frac{i}{i+x_i}$, where the $x_i$'s are distinct in $\{1,2,…,n\}$

        What is the tight upper bound of $\sum_{i=1}^n \frac{i}{i+x_i}$, where the $x_i$'s are distinct integers in $\{1,2,...,n\}$?
        5
        votes
        0answers
        150 views

        Legendre's three-square theorem and squared norm of integer matrices

        Let $\mathbb{N}$ be the set of non-negative integers. Let $E_n$ be the set of integers which are the sum of $n$ squares. Let $F_n$ be the set of integers of the form $\Vert A \Vert^2$ with $A \in M_n(...

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