arXiv subjects

Heesung Shin

Publications and source records attributed to Heesung Shin.

23 records · Page 2Linked to original sources

arXiv2010

The $q$-tangent and $q$-secant numbers via continued fractions

It is well known that the $(-1)$-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). Recently, two distinct $q$-analogues of the latter result have been discovered by Foata and Han, and Josuat-Vergès, respectively. In this paper, we will prove some general continued fractions expansions formulae, which permits us to give a unified treatment of Josuat-Vergès' two formulae and also to derive a new $q$-analogue of the aforementioned formulae. Our approach is based on a $(p,q)$-analogue of tangent and secant numbers via continued fractions and also the generating function of permutations with respect to the quintuple statistic consisting of fixed point number, weak excedance number, crossing number, nesting number and inversion number. We also give a combinatorial proof of Josuat-Vergès' formulae by using a new linear model of derangements.

math.CO↗

arXiv2010

A refined bijection between alternating permutations and 0-1-2 increasing trees

We construct a refined bijection $ϕ$ between alternating permutations and 0-1-2 increasing trees with degree at most 2. It satisfies that the first element of alternating permutation $π$ is equal to the first vertex in $ϕ(π)$ in the postorder.

math.CO↗

arXiv2009

An involution for symmetry of hook length and part length of partitions

A {\em pointed partition} of $n$ is a pair $(λ, v)$ where $λ\vdash n$ and $v$ is a cell in its Ferrers diagram. We construct an involution on pointed partitions of $n$ exchanging "hook length" and "part length". This gives a bijective proof of a recent result of Bessenrodt and Han.

math.CO↗

arXiv2008

A New Bijection Between Forests and Parking Functions

In 1980, G. Kreweras gave a recursive bijection between forests and parking functions. In this paper we construct a nonrecursive bijection from forests onto parking functions, which answers a question raised by R. Stanley. As a by-product, we obtain a bijective proof of Gessel and Seo's formula for lucky statistic on parking functions.

math.CO↗

arXiv2005

A Generalized Enumeration of Labeled Trees and Reverse Prüfer Algorithm

A {\em leader} of a tree $T$ on $[n]$ is a vertex which has no smaller descendants in $T$. Gessel and Seo showed $$\sum_{T \in \mathcal{T}_n}u^\text{(# of leaders in $T$)} c^\text{(degree of 1 in $T$)}=u P_{n-1}(1,u,cu),$$ which is a generalization of Cayley formula, where $\mathcal{T}_n$ is the set of trees on $[n]$ and $$P_n(a,b,c)=c\prod_{i=1}^{n-1}(ia+(n-i)b+c).$$ Using a variation of Prüfer code which is called a {\em RP-code}, we give a simple bijective proof of Gessel and Seo's formula.

math.CO↗