Related papers: Chess tableaux, powers of two and affine Lie algeb…

For any partition of a positive integer we consider the chess (or draughts) colouring of its associated Ferrers graph. Let b denote the total number of black unit squares, and w the number of white squares. In this note we characterize all…

Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2^[n/2].…

The number of standard Young tableaux possible of shape corresponding to 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…

A fundamental identity in the representation theory of the partition algeba is $n^k = \sum_{\lambda} f^\lambda m_k^\lambda$ for $n \geq 2k$, where $\lambda$ ranges over integer partitions of $n$, $f^\lambda$ is the number of standard Young…

A standard barely set-valued tableau of shape $\lambda$ is a filling of the Young diagram $\lambda$ with integers $1,2,\dots,|\lambda|+1$ such that the integers are increasing in each row and column, and every cell contains one integer…

Let $B(m, n)$ be the number of ways to colour a $2m \times 2n$ grid in black and white so that, in each row and each column, half of the cells are white and half are black. Bhattacharya conjectured that the exponent of $2$ in the prime…

A fundamental identity in the representation theory of the partition algebra is $n^k = \sum_{\lambda} f^\lambda m_k^\lambda$ for $n \geq 2k$, where $\lambda$ ranges over integer partitions of $n$, $f^\lambda$ is the number of standard Young…

We introduce and study an affine analogue of skew Young diagrams and tableaux on them. It turns out that the double affine Hecke algebra of type $A$ acts on the space spanned by standard tableaux on each diagram. It is shown that the…

A $d$-dimensional generalization of a Latin square of order $n$ can be considered as a chess-board of size $n\times n\times \ldots\times n$ ($d$ times), containing $n^d$ cells with $n^{d-1}$ non-attacking rooks. Each cell is identified by a…

We show that the subgraph induced in Young's graph by the set of partitions with an odd number of standard Young tableaux is a binary tree. This tree exhibits self-similarities at all scales, and has a simple recursive description.

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…

For $\ell \geq 1$ and $k \geq 2$, we consider certain admissible sequences of $k-1$ lattice paths in a colored $\ell \times \ell$ square. We show that the number of such admissible sequences of lattice paths is given by the sum of squares…

We say that a chessboard filled with integer entries satisfies the neighbour-sum property if the number appearing on each cell is the sum of entries in its neighbouring cells, where neighbours are cells sharing a common edge or vertex. We…

We consider a new kind of straight and shifted plane partitions/Young tableaux --- ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the…

We study the subgraph of the Young-Fibonacci graph induced by elements with odd $f$-statistic (the $f$-statistic of an element $w$ of a differential graded poset is the number of saturated chains from the minimal element of the poset to…

Using Symbolic Computation with Maple, we can discover lots of (rigorously-proved!) facts about Standard Young Tableaux, in particular the distribution of the entries in any specific cell, and the sorting probabilities.

We give explicit formulas for the recently introduced Schur multiple zeta values, which generalize multiple zeta(-star) values and which assign to a Young tableaux a real number. In this note we consider Young tableaux of various shapes,…

Set-valued tableaux, introduced by Buch to express the tableaux-sum formula for stable Grothendieck polynomials, generalize semistandard tableaux. We provide a new recursive proof that the number of set-valued tableaux of a given shape is…

A Latin tableau of shape $\lambda$ and type $\mu$ is a Young diagram of shape $\lambda$ in which each box contains a single positive integer, with no repeated integers in any row or column, and the $i$th most common integer appearing…

We consider the decomposition into irreducible components of the external power $\Lambda^p(\mathbb{C}^m\otimes \mathbb{C}^n)$ regarded as a $\operatorname{GL}_m\times\operatorname{GL}_n$-module. Skew Howe duality implies that the Young…