Related papers: Counting Standard Young Tableaux With Restricted R…
We prove a $q$-analog of the following result due to McKay, Morse and Wilf: the probability that a random standard Young tableau of size $n$ contains a fixed standard Young tableau of shape $\lambda\vdash k$ tends to $f^{\lambda}/k!$ in the…
An inverted semistandard Young tableau is a row-standard tableau along with a collection of inversion pairs that quantify how far the tableau is from being column semistandard. Such a tableau with precisely $k$ inversion pairs is said to be…
Young tableaux are classical combinatorial objects playing recurring and varied roles in representation theory, algebraic geometry and commutative algebra. This article is a short exposition on Young tableaux, written for the "WHAT IS...?"…
The number of standard Young tableaux of a fixed shape is famously given by the hook-length formula due to Frame, Robinson and Thrall. A bijective proof of Novelli, Pak and Stoyanovskii relies on a sorting algorithm akin to jeu-de-taquin…
Let $\mathcal{T}_3$ be the three-rowed strip. Recently Regev conjectured that the number of standard Young tableaux with $n-3$ entries in the "skew three-rowed strip" $\mathcal{T}_3 / (2,1,0)$ is $m_{n-1}-m_{n-3}$, a difference of two…
We present a generating function and a closed counting formula in two variables that enumerate a family of classes of permutations that avoid or contain an increasing pattern of length three and have a prescribed number of occurrences of…
The hook length formula gives a product formula for the number of standard Young tableaux of a partition shape. The number of standard Young tableaux of a skew shape does not always have a product formula. However, for some special skew…
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.
The $k$-Young lattice $Y^k$ is a partial order on partitions with no part larger than $k$. This weak subposet of the Young lattice originated from the study of the $k$-Schur functions(atoms) $s_\lambda^{(k)}$, symmetric functions that form…
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…
Standard set-valued Young tableaux are a generalization of standard Young tableaux where cells can contain unordered sets of integers, with the added condition that every integer at position $(i,j)$ must be smaller that every integer at…
Our main result is a limit shape theorem for the two-dimensional surface defined by a uniform random n-by-n square Young tableau. The analysis leads to a calculus of variations minimization problem that resembles the minimization problems…
We present a bijection between the set of standard Young tableaux of staircase minus rectangle shape, and the set of marked shifted standard Young tableaux of a certain shifted shape. Numerically, this result is due to DeWitt (2012).…
We define a class L_{n, k} of permutations that generalizes alternating (up-down) permutations and give bijective proofs of certain pattern-avoidance results for this class. As a special case of our results, we give two bijections between…
We characterise those classes of permutations having the property that for every tableau shape either every permutation of that shape or no permutation of that shape belongs to the class. The characterisation is in terms of the dominance…
The number of standard Young tableaux of a skew shape $\lambda/\mu$ can be computed as a sum over excited diagrams inside $\lambda$. Excited diagrams are in bijection with certain lozenge tilings, with flagged semistandard tableaux and also…
Inspired by work of McKay, Morse, and Wilf, we give an exact count of the involutions in S_n which contain a given permutation \tau in S_k as a subsequence; this number depends on the patterns of the first j values of \tau for 1<=j<=k. We…
A tableau inversion is a pair of entries from the same column of a row-standard tableau that lack the relative ordering necessary to make the tableau column-standard. An $i$-inverted Young tableau is a row-standard tableau with precisely…
We prove that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles. We show that this is best-possible by partitioning…
New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…