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We consider the symmetric group $S_n$-module of the polynomial ring with $m$ sets of $n$ commuting variables and $m'$ sets of $n$ anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition $\lambda$…

It is known that a graph isomorphism testing algorithm is polynomially equivalent to a detecting of a graph non-trivial automorphism algorithm. The polynomiality of the latter algorithm, is obtained by consideration of symmetry properties…

The paper introduces the notion of a representation $k$-graph $(\Delta,\alpha)$ for a given $k$-graph $\Lambda$. It is shown that any representation $k$-graph for $\Lambda$ yields a module for the Kumjian-Pask algebra $KP(\Lambda)$, and the…

In \cite{[CZ]}, Cohen and Zemel showed that for a partition $\lambda \vdash k$, the dimension of the irreducible representation of $S_{n}$ corresponding to the partition $(n-k,\lambda) \vdash n$ is a polynomial of degree $k$ in $n$, whose…

In this article we use existing machinery to define connective $K$-theory spectra associated to topological ringoids. Algebraic $K$-theory of discrete ringoids, and the analytic $K$-theory of Banach categories are obtained as special cases.…

In this paper we discuss the "Factorization phenomenon" which occurs when a representation of a Lie algebra is restricted to a subalgebra, and the result factors into a tensor product of smaller representations of the subalgebra. We analyze…

In this paper we prove a duality between $k$-noncrossing partitions over $[n]=\{1,...,n\}$ and $k$-noncrossing braids over $[n-1]$. This duality is derived directly via (generalized) vacillating tableaux which are in correspondence to…

We give three proofs of the following result conjectured by Carriegos, De Castro-Garc\'{\i}a and Mu\~noz Casta\~neda in their work on enumeration of control systems: when $\binom{k+1}{2} \le n < \binom{k+2}{2}$, there are as many partitions…

We propound the thesis that there is a limitation to the number of possible structures which are axiomatically endowed with identities involving operations. In the case of algebras with a binary operation satisfying a formally reducible (to…

We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the dimer partition…

Ballantine--Beck--Feigon--Maurischat introduced the subsum polynomial \[ \operatorname{sp}(\lambda,x):=\prod_i (1+x^{\lambda_i}) \] attached to an integer partition $\lambda$, and studied rational functions obtained by summing reciprocals…

We prove an identity about partitions, previously conjectured in the study of shifted Jack polynomials (math.CO/9903020). The proof given is using $\lambda$-ring techniques. It would be interesting to obtain a bijective proof.

Let $a_k(n)$ denotes the number of representations of a non-negative integer $n$ as sum of $k$ quadratic forms of the type $x^2+xy+y^2$ and $a_{\lambda_1,\lambda_2,\lambda_3\dots\lambda_k}(n)$ denotes the number of representations $n$ as a…

The conjugacy classes of nilpotent $n\times n$ matrices can be parametrised by partitions $\lambda$ of $n$, and for a nilpotent $\eta$ in the class parametrised by $\lambda$, the variety $F_\eta$ of $\eta$-stable flags has its irreducible…

This paper defines the partition algebra for complex reflection group $G(r,p,n)$ acting on $k$-fold tensor product $(\mathbb{C}^n)^{\otimes k}$, where $\mathbb{C}^n$ is the reflection representation of $G(r,p,n)$. A basis of the centralizer…

There is a classical connection between the representation theory of the symmetric group and the general linear group called Schur-Weyl duality. Variations on this principle yield analogous connections between the symmetric group and other…

The number of standard Young tableaux of shape a partition $\lambda$ is called the dimension of the partition and is denoted by $f^{\lambda}$. Partitions with odd dimensions were enumerated by McKay and were further characterized by…

If $[\lambda(j)]$ is a multipartition of the positive integer $n$ (a sequence of partitions with total size $n$), and $\mu$ is a partition of $n$, we study the number $K_{[\lambda(j)]\mu}$ of sequences of semistandard Young tableaux of…

If $\gcd(r,t)=1$, then a theorem of Alladi offers the M\"obius sum identity $$-\sum_{\substack{ n \geq 2 \\ p_{\rm{min}}(n) \equiv r \pmod{t}}} \mu(n)n^{-1}= \frac{1}{\varphi(t)}. $$ Here $p_{\rm{min}}(n)$ is the smallest prime divisor of…

We introduce the dual affine Robinson-Schensted correspondence that gives a bijection between the extended affine symmetric group and tuples $(\bar{P},\bar{Q},\lambda,N)$, where $\bar{P}$ and $\bar{Q}$ are tabloids, $\lambda$ is a…