scholarly journals Combinatorics of Tableau Inversions

10.37236/4932 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Jonathan E. Beagley ◽  
Paul Drube
Keyword(s):  
Betti Number ◽  
Young Tableau ◽  
Betti Numbers ◽  
Young Tableaux ◽  
Standard Tableau ◽  
Specific Standard ◽  
Different Shapes ◽  
Springer Fibers ◽  
Standard Young Tableau ◽  
Fixed Standard

A tableau inversion is a pair of entries in row-standard tableau $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard.  An $i$-inverted Young tableau is a row-standard tableau along with precisely $i$ inversion pairs. Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of $i$-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableau inversions from a completely combinatorial perspective. We develop formulas enumerating the number of $i$-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableau that standardize a specific standard Young tableau, and construct bijections between $i$-inverted Young tableaux of a certain shape with $j$-inverted Young tableaux of different shapes. Finally, we share some the results of a computer program developed to calculate tableaux inversions.

scholarly journals Inversions of Semistandard Young Tableaux

10.37236/5469 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Paul Drube
Keyword(s):  
Young Tableau ◽  
Betti Numbers ◽  
Young Tableaux ◽  
Closed Formula ◽  
General Shape ◽  
Combinatorial Objects ◽  
Standard Tableau ◽  
Specific Shape ◽  
Springer Fibers ◽  
Standard Young Tableaux

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 $i$ inversion pairs, and may be interpreted as a generalization of (column-standard) Young tableaux. Inverted Young tableaux that lack repeated entries were introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, and were later developed as combinatorial objects in their own right by Beagley and Drube. This paper generalizes earlier notions of tableau inversions to row-standard tableaux with repeated entries, yielding an interesting new generalization of semistandard (as opposed to merely standard) Young tableaux. We develop a closed formula for the maximum numbers of inversion pairs for a row-standard tableau with a specific shape and content, and show that the number of $i$-inverted tableaux of a given shape is invariant under permutation of content. We then enumerate $i$-inverted Young tableaux for a variety of shapes and contents, and generalize an earlier result that places $1$-inverted Young tableaux of a general shape in bijection with $0$-inverted Young tableaux of a variety of related shapes.

scholarly journals Asymptotics for the Distributions of Subtableaux in Young and Up-Down Tableaux

10.37236/1886 ◽  
2006 ◽  
Vol 11 (2) ◽  
Author(s):  
David J. Grabiner
Keyword(s):  
Random Walk ◽  
Young Tableau ◽  
Young Tableaux ◽  
Similar Formula ◽  
K Cells ◽  
Standard Tableau ◽  
Asymptotic Probability ◽  
Standard Young Tableau

Let $\mu$ be a partition of $k$, and $T$ a standard Young tableau of shape $\mu$. McKay, Morse, and Wilf show that the probability a randomly chosen Young tableau of $N$ cells contains $T$ as a subtableau is asymptotic to $f^\mu/k!$ as $N$ goes to infinity, where $f^\mu$ is the number of all tableaux of shape $\mu$. We use a random-walk argument to show that the analogous asymptotic probability for randomly chosen Young tableaux with at most $n$ rows is proportional to $\prod_{1\le i < j\le n}\bigl((\mu_i-i)-(\mu_j-j)\bigr)$; as $n$ goes to infinity, the probabilities approach $f^\mu/k!$ as expected. We have a similar formula for up-down tableaux; the probability approaches $f^\mu/k!$ if $\mu$ has $k$ cells and thus the up-down tableau is actually a standard tableau, and approaches 0 if $\mu$ has fewer than $k$ cells.

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 Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers

10.37236/6376 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Paul Drube
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Young Tableau ◽  
Young Tableaux ◽  
Dyck Paths ◽  
Standard Tableau ◽  
Ballot Numbers ◽  
Standard Young Tableaux

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 a $k$-inverted semistandard Young tableau. Building upon earlier work by Fresse and the author, this paper develops generating functions for the numbers of $k$-inverted semistandard Young tableaux of various shapes $\lambda$ and contents $\mu$. An easily-calculable generating function is given for the number of $k$-inverted semistandard Young tableaux that "standardize" to a fixed semistandard Young tableau. For $m$-row shapes $\lambda$ and standard content $\mu$, the total number of $k$-inverted standard Young tableaux of shape $\lambda$ is then enumerated by relating such tableaux to $m$-dimensional generalizations of Dyck paths and counting the numbers of "returns to ground" in those paths. In the rectangular specialization of $\lambda = n^m$ this yields a generating function that involves $m$-dimensional analogues of the famed Ballot numbers. Our various results are then used to directly enumerate all $k$-inverted semistandard Young tableaux with arbitrary content and two-row shape $\lambda = a^1 b^1$, as well as all $k$-inverted standard Young tableaux with two-column shape $\lambda=2^n$.

scholarly journals A Sundaram type Bijection for SO(3): Vacillating Tableaux and Pairs of Standard Young Tableaux and Orthogonal Littlewood-Richardson Tableaux

10.37236/7713 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Judith Jagenteufel
Keyword(s):  
Orthogonal Group ◽  
Young Tableau ◽  
Young Tableaux ◽  
Tensor Power ◽  
Direct Sum ◽  
Special Orthogonal Group ◽  
Direct Sum Decomposition ◽  
Descent Set ◽  
Standard Young Tableaux ◽  
Standard Young Tableau

Motivated by the direct-sum-decomposition of the $r^{\text{th}}$ tensor power of the defining representation of the special orthogonal group $\mathrm{SO}(2k + 1)$, we present a bijection between vacillating tableaux and pairs consisting of a standard Young tableau and an orthogonal Littlewood-Richardson tableau for $\mathrm{SO}(3)$.Our bijection preserves a suitably defined descent set. Using it we determine the quasi-symmetric expansion of the Frobenius characters of the isotypic components.On the combinatorial side we obtain a bijection between Riordan paths and standard Young tableaux with 3 rows, all of even length or all of odd length.

scholarly journals Symmetry properties of the Novelli-Pak-Stoyanovskii algorithm

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Robin Sulzgruber
Keyword(s):  
Young Tableau ◽  
Sorting Algorithm ◽  
Young Tableaux ◽  
Hook Length ◽  
Jeu De Taquin ◽  
Bijective Proof ◽  
Symmetry Properties ◽  
International Audience ◽  
Standard Young Tableaux ◽  
Standard Young Tableau

International audience 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 which transforms an arbitrary filling of a partition into a standard Young tableau by exchanging adjacent entries. Recently, Krattenthaler and Müller defined the complexity of this algorithm as the average number of performed exchanges, and Neumann and the author proved it fulfils some nice symmetry properties. In this paper we recall and extend the previous results and provide new bijective proofs.

A maxdrop statistic for standard Young tableaux

2021 ◽  
pp. 2150105
Author(s):  
Mark Dukes ◽  
Toufik Mansour
Keyword(s):  
Generating Function ◽  
Young Tableau ◽  
Young Tableaux ◽  
Standard Young Tableaux ◽  
Permutation Statistic ◽  
Standard Young Tableau

In this paper, we introduce a new statistic on standard Young tableaux that is closely related to the maxdrop permutation statistic that was introduced by the first author. We prove that the value of the statistic must be attained at one of the corners of the standard Young tableau. We determine the coefficients of the generating function of this statistic over two-row standard Young tableaux having [Formula: see text] cells. We prove several results for this new statistic that include unimodality of the coefficients for the two-row case.

scholarly journals Prym–Brill–Noether Loci of Special Curves

2020 ◽  
Author(s):  
Steven Creech ◽  
Yoav Len ◽  
Caelan Ritter ◽  
Derek Wu
Keyword(s):  
Upper Bound ◽  
Betti Number ◽  
Young Tableaux

Abstract We use Young tableaux to compute the dimension of $V^r$, the Prym–Brill–Noether locus of a folded chain of loops of any gonality. This tropical result yields a new upper bound on the dimensions of algebraic Prym–Brill–Noether loci. Moreover, we prove that $V^r$ is pure dimensional and connected in codimension $1$ when $\dim V^r \geq 1$. We then compute the 1st Betti number of this locus for even gonality when the dimension is exactly $1$ and compute the cardinality when the locus is finite and the edge lengths are generic.

scholarly journals APPROXIMATING THE FIRST L2-BETTI NUMBER OF RESIDUALLY FINITE GROUPS

2011 ◽  
Vol 03 (02) ◽  
pp. 153-160 ◽  
Author(s):  
W. LÜCK ◽  
D. OSIN
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Betti Number ◽  
Betti Numbers ◽  
Torsion Group ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Residually Finite Groups

We show that the first L2-betti number of a finitely generated residually finite group can be estimated from below by using ordinary first betti numbers of finite index normal subgroups. As an application, we construct a finitely generated infinite residually finite torsion group with positive first L2-betti number.

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