scholarly journals On the Sweep Map for $\vec{k}$-Dyck Paths

10.37236/8346 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Guoce Xin ◽  
Yingrui Zhang
Keyword(s):  
Similar Type ◽  
Young Tableaux ◽  
Linear Algorithm ◽  
Dyck Paths ◽  
Standard Young Tableaux ◽  
Parameter Case

Garsia and Xin gave a linear algorithm for inverting the sweep map for Fuss rational Dyck paths in $D_{m,n}$ where $m=kn\pm 1$. They introduced an intermediate family $\mathcal{T}_n^k$ of certain standard Young tableaux. Then inverting the sweep map is done by a simple walking algorithm on a $T\in \mathcal{T}_n^k$. We find their idea naturally extends for $\mathbf{k}^\pm$-Dyck paths, and also for $\mathbf{k}$-Dyck paths (reducing to $k$-Dyck paths for the equal parameter case). The intermediate object becomes a similar type of tableau in $\mathcal{T}_\mathbf{k}$ of different column lengths. This approach is independent of the Thomas-Williams algorithm for inverting the general modular sweep map.

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 From Dyck Paths to Standard Young Tableaux

2020 ◽  
Vol 24 (1) ◽  
pp. 69-93
Author(s):  
Juan B. Gil ◽  
Peter R. W. McNamara ◽  
Jordan O. Tirrell ◽  
Michael D. Weiner
Keyword(s):  
Young Tableaux ◽  
Dyck Paths ◽  
Standard Young Tableaux

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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