scholarly journals Towards the Distribution of the Size of a Largest Planar Matching and Largest Planar Subgraph in Random Bipartite Graphs

10.37236/859 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Marcos Kiwi ◽  
Martin Loebl
Keyword(s):  
Generating Function ◽  
Maximum Size ◽  
Pattern Avoidance ◽  
Combinatorial Identities ◽  
Young Tableaux ◽  
Color Class ◽  
Lattice Walks ◽  
Type Property ◽  
Bivariate Generating Function ◽  
Standard Young Tableaux

We address the following question: When a randomly chosen regular bipartite multi–graph is drawn in the plane in the "standard way", what is the distribution of its maximum size planar matching (set of non–crossing disjoint edges) and maximum size planar subgraph (set of non–crossing edges which may share endpoints)? The problem is a generalization of the Longest Increasing Sequence (LIS) problem (also called Ulam's problem). We present combinatorial identities which relate the number of $r$-regular bipartite multi–graphs with maximum planar matching (maximum planar subgraph) of at most $d$ edges to a signed sum of restricted lattice walks in ${\Bbb Z}^d$, and to the number of pairs of standard Young tableaux of the same shape and with a "descend–type" property. Our results are derived via generalizations of two combinatorial proofs through which Gessel's identity can be obtained (an identity that is crucial in the derivation of a bivariate generating function associated to the distribution of the length of LISs, and key to the analytic attack on Ulam's problem). Finally, we generalize Gessel's identity. This enables us to count, for small values of $d$ and $r$, the number of $r$-regular bipartite multi-graphs on $n$ nodes per color class with maximum planar matchings of size $d$.Our work can also be viewed as a first step in the study of pattern avoidance in ordered bipartite multi-graphs.

scholarly journals Pattern avoidance in alternating permutations and tableaux (extended abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Joel Brewster Lewis
Keyword(s):  
Pattern Avoidance ◽  
Catalan Numbers ◽  
Young Tableaux ◽  
Alternating Permutations ◽  
Special Cases ◽  
Nous Montrons ◽  
Insertion Algorithm ◽  
Generating Trees ◽  
International Audience ◽  
Standard Young Tableaux

International audience We give bijective proofs of pattern-avoidance results for a class of permutations generalizing alternating permutations. The bijections employed include a modified form of the RSK insertion algorithm and recursive bijections based on generating trees. As special cases, we show that the sets $A_{2n}(1234)$ and $A_{2n}(2143)$ are in bijection with standard Young tableaux of shape $\langle 3^n \rangle$. Alternating permutations may be viewed as the reading words of standard Young tableaux of a certain skew shape. In the last section of the paper, we study pattern avoidance in the reading words of standard Young tableaux of any skew shape. We show bijectively that the number of standard Young tableaux of shape $\lambda / \mu$ whose reading words avoid $213$ is a natural $\mu$-analogue of the Catalan numbers. Similar results for the patterns $132$, $231$ and $312$. Nous présentons des preuves bijectives de résultats pour une classe de permutations à motifs exclus qui généralisent les permutations alternantes. Les bijections utilisées reposent sur une modification de l'algorithme d'insertion "RSK" et des bijections récursives basées sur des arbres de génération. Comme cas particuliers, nous montrons que les ensembles $A_{2n}(1234)$ et $A_{2n}(2143)$ sont en bijection avec les tableaux standards de Young de la forme $\langle 3^n \rangle$. Une permutation alternante peut être considérée comme le mot de lecture de certain skew tableau. Dans la dernière section de l'article, nous étudions l'évitement des motifs dans les mots de lecture de skew tableaux généraux. Nous montrons bijectivement que le nombre de tableaux standards de forme $\lambda / \mu$ dont les mots de lecture évitent $213$ est un $\mu$-analogue naturel des nombres de Catalan. Des résultats analogues sont valables pour les motifs $132$, $231$ et $312$.

scholarly journals Generating Trees and Pattern Avoidance in Alternating Permutations

10.37236/1173 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Joel Brewster Lewis
Keyword(s):  
Pattern Avoidance ◽  
Young Tableaux ◽  
Alternating Permutations ◽  
Open Questions ◽  
Permutation Pattern ◽  
Generating Trees ◽  
Generating Tree ◽  
Standard Young Tableaux ◽  
Tree Approach

We extend earlier work of the same author to enumerate alternating permutations avoiding the permutation pattern $2143$.  We use a generating tree approach to construct a recursive bijection between the set $A_{2n}(2143)$ of alternating permutations of length $2n$ avoiding $2143$ and the set of standard Young tableaux of shape $\langle n, n, n\rangle$, and between the set $A_{2n + 1}(2143)$ of alternating permutations of length $2n + 1$ avoiding $2143$ and the set of shifted standard Young tableaux of shape $\langle n + 2, n + 1, n\rangle$.  We also give a number of conjectures and open questions on pattern avoidance in alternating permutations and generalizations thereof.

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 Kirillov's Unimodality Conjecture for the Rectangular Narayana Polynomials

10.37236/6806 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Herman Z. Q. Chen ◽  
Arthur L. B. Yang ◽  
Philip B. Zhang
Keyword(s):  
Generating Function ◽  
Catalan Numbers ◽  
Young Tableaux ◽  
Rectangular Shape ◽  
The Real ◽  
Real Zeros ◽  
Kostka Numbers ◽  
Standard Young Tableaux

In the study of Kostka numbers and Catalan numbers, Kirillov posed a unimodality conjecture for the rectangular Narayana polynomials. We prove that the rectangular Narayana polynomials have only real zeros, and thereby confirm Kirillov's unimodality conjecture. By using an equidistribution property between descent numbers and ascent numbers on ballot paths due to Sulanke and a bijection between lattice words and standard Young tableaux, we show that the rectangular Narayana polynomial is equal to the descent generating function on standard Young tableaux of certain rectangular shape, up to a power of the indeterminate. Then we obtain the real-rootedness of the rectangular Narayana polynomial based on a result of Brenti which implies that the descent generating function of standard Young tableaux has only real zeros.

scholarly journals Subsequence Containment by Involutions

10.37236/1911 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Aaron D. Jaggard
Keyword(s):  
Pattern Avoidance ◽  
Young Tableaux ◽  
Standard Young Tableaux

Inspired by work of McKay, Morse, and Wilf, we give an exact count of the involutions in ${\cal S}_{n}$ which contain a given permutation $\tau\in{\cal S}_{k}$ as a subsequence; this number depends on the patterns of the first $j$ values of $\tau$ for $1\leq j\leq k$. We then use this to define a partition of ${\cal S}_{k}$, analogous to Wilf-classes in the study of pattern avoidance, and examine properties of this equivalence. In the process, we show that a permutation $\tau_1\ldots\tau_k$ is layered iff, for $1\leq j\leq k$, the pattern of $\tau_1\ldots\tau_j$ is an involution. We also obtain a result of Sagan and Stanley counting the standard Young tableaux of size $n$ which contain a fixed tableau of size $k$ as a subtableau.

scholarly journals Pattern Avoidance and Young Tableaux

10.37236/6427 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Zhousheng Mei ◽  
Suijie Wang
Keyword(s):  
General Class ◽  
Pattern Avoidance ◽  
Catalan Numbers ◽  
Young Tableaux ◽  
Standard Young Tableaux

This paper extends Lewis's bijection (J. Combin. Theorey Ser. A 118, 2011) to a bijection between a more general class $\mathcal{L}(n,k,I)$ of permutations and the set of standard Young tableaux of shape $\langle (k+1)^n\rangle$, so the cardinality\[|\mathcal{L}(n,k,I)|=f^{\langle (k+1)^n\rangle},\]is independent of the choice of $I\subseteq [n]$. As a consequence, we obtain some new combinatorial realizations and identities on Catalan numbers. In the end, we raise a problem on finding a bijection between $\mathcal{L}(n,k,I)$ and $\mathcal{L}(n,k,I')$ for distinct $I$ and $I'$.

scholarly journals A Relationship between the Major Index for Tableaux and the Charge Statistic for Permutations

10.37236/1942 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Kendra Killpatrick
Keyword(s):  
Linear Group ◽  
Generating Function ◽  
General Linear Group ◽  
General Linear ◽  
Combinatorial Proof ◽  
Young Tableaux ◽  
Major Index ◽  
Combinatorial Interpretations ◽  
Standard Young Tableaux ◽  
Kostka Number

The widely studied $q$-polynomial $f^{\lambda}(q)$, which specializes when $q=1$ to $f^{\lambda}$, the number of standard Young tableaux of shape $\lambda$, has multiple combinatorial interpretations. It represents the dimension of the unipotent representation $S_q^{\lambda}$ of the finite general linear group $GL_n(q)$, it occurs as a special case of the Kostka-Foulkes polynomials, and it gives the generating function for the major index statistic on standard Young tableaux. Similarly, the $q$-polynomial $g^{\lambda}(q)$ has combinatorial interpretations as the $q$-multinomial coefficient, as the dimension of the permutation representation $M_q^{\lambda}$ of the general linear group $GL_n(q)$, and as the generating function for both the inversion statistic and the charge statistic on permutations in $W_{\lambda}$. It is a well known result that for $\lambda$ a partition of $n$, $dim(M_q^{\lambda}) = \Sigma_{\mu} K_{\mu \lambda} dim(S_q^{\mu})$, where the sum is over all partitions $\mu$ of $n$ and where the Kostka number $K_{\mu \lambda}$ gives the number of semistandard Young tableaux of shape $\mu$ and content $\lambda$. Thus $g^{\lambda}(q) - f^{\lambda}(q)$ is a $q$-polynomial with nonnegative coefficients. This paper gives a combinatorial proof of this result by defining an injection $f$ from the set of standard Young tableaux of shape $\lambda$, $SYT(\lambda)$, to $W_{\lambda}$ such that $maj(T) = ch(f(T))$ for $T \in SYT(\lambda)$.

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.

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
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