Estimate of Time Needed for a Coordinate of a Bernoulli Scheme to Fall into the First Column of a Young Tableau

2020 ◽  
Vol 54 (2) ◽  
pp. 135-140
Author(s):  
I. F. Azangulov ◽  
G. V. Ovechkin
Keyword(s):  
Young Tableau ◽  
Bernoulli Scheme

scholarly journals Dual Creation Operators and a Dendriform Algebra Structure on the Quasisymmetric Functions

2017 ◽  
Vol 69 (1) ◽  
pp. 21-53 ◽  
Author(s):  
Darij Grinberg
Keyword(s):  
Hopf Algebras ◽  
Young Tableau ◽  
Vertex Operators ◽  
Schur Function ◽  
Algebra Structure ◽  
Quasisymmetric Functions ◽  
Combinatorial Hopf Algebras ◽  
Dendriform Algebra ◽  
Natural Analogues ◽  
Creation Operators

AbstractThe dual immaculate functions are a basis of the ring QSym of quasisymmetric functions and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an immaculate tableau is defined similarly to a semistandard Young tableau, but the shape is a composition rather than a partition, and only the first column is required to strictly increase (whereas the other columns can be arbitrary, but each row has to weakly increase). Dual immaculate functions were introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki in arXiv:1208.5191, and have since been found to possess numerous nontrivial properties.In this note, we prove a conjecture of M. Zabrocki that provides an alternative construction for the dual immaculate functions in terms of certain “vertex operators”. The proof uses a dendriform structure on the ring QSym; we discuss the relation of this structure to known dendriformstructures on the combinatorial Hopf algebras FQSym andWQSym.

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 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.

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