scholarly journals A Note on Statistical Averages for Oscillating Tableaux

10.37236/4641 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Sam Hopkins ◽  
Ingrid Zhang
Keyword(s):  
Simple Formula ◽  
Perfect Matchings ◽  
Quadratic Polynomial ◽  
Average Weight ◽  
Young Tableaux ◽  
Hidden Symmetry ◽  
Conceptual Proof ◽  
Standard Young Tableaux ◽  
Young’S Lattice

Oscillating tableaux are certain walks in Young's lattice of partitions; they generalize standard Young tableaux. The shape of an oscillating tableau is the last partition it visits and the length of an oscillating tableau is the number of steps it takes. We define a new statistic for oscillating tableaux that we call weight: the weight of an oscillating tableau is the sum of the sizes of all the partitions that it visits.  We show that the average weight of all oscillating tableaux of shape $\lambda$ and length $|\lambda|+2n$ (where $|\lambda|$ denotes the size of $\lambda$ and $n \in \mathbb{N}$) has a surprisingly simple formula: it is a quadratic polynomial in $|\lambda|$ and $n$. Our proof via the theory of differential posets is largely computational. We suggest how the homomesy paradigm of Propp and Roby may lead to a more conceptual proof of this result and reveal a hidden symmetry in the set of perfect matchings.

scholarly journals Rook Placements and Jordan Forms of Upper-Triangular Nilpotent Matrices

10.37236/6888 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Martha Yip
Keyword(s):  
Finite Field ◽  
Weighted Sum ◽  
Young Tableaux ◽  
Canonical Forms ◽  
Combinatorial Formula ◽  
Rook Placements ◽  
Nilpotent Matrices ◽  
Jordan Forms ◽  
Standard Young Tableaux ◽  
Young’S Lattice

The set of $n$ by $n$ upper-triangular nilpotent matrices with entries in a finite field $\mathbb{F}_q$ has Jordan canonical forms indexed by partitions $\lambda \vdash n$. We present a combinatorial formula for computing the number $F_\lambda(q)$ of matrices of Jordan type $\lambda$ as a weighted sum over standard Young tableaux. We construct a bijection between paths in a modified version of Young's lattice and non-attacking rook placements, which leads to a refinement of the formula for $F_\lambda(q)$.

scholarly journals Homomesy in Products of Two Chains

10.37236/3579 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
James Propp ◽  
Tom Roby
Keyword(s):  
Theoretical Framework ◽  
Vector Spaces ◽  
Young Tableaux ◽  
Combinatorial Objects ◽  
Standard Young Tableaux ◽  
Young’S Lattice

Many invertible actions $\tau$ on a set $\mathcal{S}$ of combinatorial objects, along with a natural statistic $f$ on $\mathcal{S}$, exhibit the following property which we dub homomesy: the average of $f$ over each $\tau$-orbit in $\mathcal{S}$ is the same as the average of $f$ over the whole set $\mathcal{S}$. This phenomenon was first noticed by Panyushev in 2007 in the context of the rowmotion action on the set of antichains of a root poset; Armstrong, Stump, and Thomas proved Panyushev's conjecture in 2011. We describe a theoretical framework for results of this kind  that applies more broadly, giving examples in a variety of contexts. These include linear actions on vector spaces, sandpile dynamics, Suter's action on certain subposets of Young's Lattice, Lyness 5-cycles, promotion of rectangular semi-standard Young tableaux, and the rowmotion and promotion actions on certain posets. We give a detailed description of the latter situation for products of two chains.  

Standard Young tableaux in a (2,1)-hook and Motzkin paths

2021 ◽  
Vol 344 (7) ◽  
pp. 112395
Author(s):  
Rosena R.X. Du ◽  
Jingni Yu
Keyword(s):  
Young Tableaux ◽  
Standard Young Tableaux ◽  
Motzkin Paths

scholarly journals A direct bijective proof of the hook-length formula

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Jean-Christophe Novelli ◽  
Igor Pak ◽  
Alexander V. Stoyanovskii
Keyword(s):  
Young Tableaux ◽  
Hook Length ◽  
Bijective Proof ◽  
International Audience ◽  
Standard Young Tableaux

International audience This paper presents a new proof of the hook-length formula, which computes the number of standard Young tableaux of a given shape. After recalling the basic definitions, we present two inverse algorithms giving the desired bijection. The next part of the paper presents the proof of the bijectivity of our construction. The paper concludes with some examples.

scholarly journals Enumeration of Standard Young Tableaux of Shifted Strips with Constant Width

10.37236/6466 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Ping Sun
Keyword(s):  
Integral Method ◽  
Recurrence Relations ◽  
Young Tableau ◽  
Constant Width ◽  
Young Tableaux ◽  
Pell Number ◽  
Standard Young Tableaux ◽  
Standard Young Tableau

Let $g_{n_1,n_2}$ be the number of standard Young tableau of truncated shifted shape with $n_1$ rows and $n_2$ boxes in each row. By using the integral method this paper derives the recurrence relations of $g_{3,n}$, $g_{n,4}$ and $g_{n,5}$ respectively. Specifically, $g_{n,4}$ is the $(2n-1)$-st Pell number.

scholarly journals Evaluating the Numbers of some Skew Standard Young Tableaux of Truncated Shapes

10.37236/3890 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Ping Sun
Keyword(s):  
Young Tableaux ◽  
Multiple Integrals ◽  
Standard Young Tableaux ◽  
Product Formulas

In this paper the number of standard Young tableaux (SYT) is evaluated by the methods of multiple integrals and combinatorial summations. We obtain the product formulas of the numbers of skew SYT of certain truncated shapes, including the skew SYT $((n+k)^{r+1},n^{m-1}) / (n-1)^r $ truncated by a rectangle or nearly a rectangle, the skew SYT of truncated shape $((n+1)^3,n^{m-2}) / (n-2) \backslash \; (2^2)$, and the SYT of truncated shape $((n+1)^2,n^{m-2}) \backslash \; (2)$.

scholarly journals On two related questions of Wilf concerning Standard Young Tableaux

2009 ◽  
Vol 30 (5) ◽  
pp. 1318-1322 ◽  
Author(s):  
Miklós Bóna
Keyword(s):  
Young Tableaux ◽  
Standard Young Tableaux

Standard Young tableaux and character generators of classical Lie groups

1984 ◽  
Vol 17 (1) ◽  
pp. 19-45 ◽  
Author(s):  
R C King ◽  
N G I El-Sharkaway
Keyword(s):  
Lie Groups ◽  
Young Tableaux ◽  
Standard Young Tableaux

scholarly journals Stack words, standard Young tableaux, permutations with forbidden subsequences and planar maps

2000 ◽  
Vol 210 (1-3) ◽  
pp. 71-85 ◽  
Author(s):  
O Guibert
Keyword(s):  
Young Tableaux ◽  
Planar Maps ◽  
Standard Young Tableaux
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