Permutation Factorizations and Prime Parking Functions

Rattan, Amarpreet

doi:10.1007/s00026-006-0285-7

Cited by 5 publications

(3 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Comparing this expression with (13), the result follows. ✷ Indeed, using Lagrange inversion we see that the linear terms of R p; q (z) and G p; q (z) are 0 and m i=0 p i z, respectively.…”

Section: Terms Of Degree K +mentioning

confidence: 85%

“…The expression for the terms of highest degree in Stanley's polynomials are given implicitly by G p; q (z) in (13). From Theorem 1.5 and Proposition 4.1, we can obtain a similar formula for the highest degree terms; that is, the terms of highest degree in F k (p; q), which have degree k + 1, is given by R k+1 (p; q) and we see that the generating series for the terms of highest degree is…”

Section: Terms Of Degree K +mentioning

confidence: 99%

See 1 more Smart Citation

Positivity results for Stanley's character polynomials

Rattan

2007

Journal of Algebra

View full text Add to dashboard Cite

In Stanley [R.P. Stanley, Irreducible symmetric group characters of rectangular shape, Sém. Lothar. Combin. 50 (B50d) (2003) 11 pp.] the author introduces expressions for the normalized characters of the symmetric group and states some positivity conjectures for these expressions. Here, we give an affirmative partial answer to Stanley's positivity conjectures about the expressions using results on Kerov polynomials. In particular, we use new positivity results in Goulden and Rattan [I.P. Goulden, A. Rattan, An explicit form for Kerov's character polynomials, Trans. Amer. Math. Soc., in press, math.CO/0505317, November 2005].We shall see that the generating series C(t) introduced in [I.P. Goulden, A. Rattan, An explicit form for Kerov's character polynomials, Trans. Amer. Math. Soc., in press, math.CO/0505317, November 2005] is critical to our discussion.

show abstract

Section: Terms Of Degree K +mentioning

confidence: 85%

Section: Terms Of Degree K +mentioning

confidence: 99%

Positivity results for Stanley's character polynomials

Rattan

2007

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…Although some enumerative aspects of parking functions have been previously studied [6,12,15], we are not aware of any previous results dealing with pattern avoidance in this setting.…”

Section: Parking Functionsmentioning

confidence: 99%

Wilf-Equivalence on $k$-ary Words, Compositions, and Parking Functions

Jelínek¹,

Mansour²

2009

Electron. J. Combin.

View full text Add to dashboard Cite

In this paper, we study pattern-avoidance in the set of words over the alphabet $[k]$. We say that a word $w\in[k]^n$ contains a pattern $\tau\in[\ell]^m$, if $w$ contains a subsequence order-isomorphic to $\tau$. This notion generalizes pattern-avoidance in permutations. We determine all the Wilf-equivalence classes of word patterns of length at most six. We also consider analogous problems within the set of integer compositions and the set of parking functions, which may both be regarded as special types of words, and which contain all permutations. In both these restricted settings, we determine the equivalence classes of all patterns of length at most five. As it turns out, the full classification of these short patterns can be obtained with only a few general bijective arguments, which are applicable to patterns of arbitrary size.

show abstract

A combinatorial interpretation of the bijection of Goulden and Yong

Ojakian

2020

AKCE International Journal of Graphs and Combinatorics

View full text Add to dashboard Cite

We define a dual of a graph, generalizing the definition of Goulden et al. (2002), which only applies to trees; then we reprove their main result using our new definition. We in fact give two characterizations of the dual: a graph-theoretic one and an algebraic one, showing that they are in fact the same, and are also the same (on trees) as the topological dual developed by Goulden and Yong. Goulden and Yong use their dual to define a bijection between the vertex labeled trees and the factorizations of the permutation (n, . . . , 1) into n − 1 transpositions, showing that their bijection has a particular structural property. We give a combinatorial, non-topological, proof of their result, using our dual.

show abstract

Permutation Factorizations and Prime Parking Functions

Cited by 5 publications

References 10 publications

Positivity results for Stanley's character polynomials

Positivity results for Stanley's character polynomials

Wilf-Equivalence on $k$-ary Words, Compositions, and Parking Functions

A combinatorial interpretation of the bijection of Goulden and Yong

Contact Info

Product

Resources

About