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Register a new userA new decomposition of ascent sequences and Euler--Stirling statistics
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As shown by Bousquet-Melou--Claesson--Dukes--Kitaev (2010), ascent sequences can be used to encode $({bf2+2})$-free posets. It is known that ascent sequences are enumerated by the Fishburn numbers, which appear as the coefficients of the formal power series $$sum_{m=1}^{infty}prod_{i=1}^m (1-(1-t)^i).$$ In this paper, we present a novel way to recursively decompose ascent sequences, which leads to: (i) a calculation of the Euler--Stirling distribution on ascent sequences, including the numbers of ascents ($asc$), repeated entries $(rep)$, zeros ($zero$) and maximal entries ($max$). In particular, this confirms and extends Dukes and Parviainens conjecture on the equidistribution of $zero$ and $max$. (ii) a far-reaching generalization of the generating function formula for $(asc,zero)$ due to Jelinek. This is accomplished via a bijective proof of the quadruple equidistribution of $(asc,rep,zero,max)$ and $(rep,asc,rmin,zero)$, where $rmin$ denotes the right-to-left minima statistic of ascent sequences. (iii) an extension of a conjecture posed by Levande, which asserts that the pair $(asc,zero)$ on ascent sequences has the same distribution as the pair $(rep,max)$ on $({bf2-1})$-avoiding inversion sequences. This is achieved via a decomposition of $({bf2-1})$-avoiding inversion sequences parallel to that of ascent sequences. This work is motivated by a double Eulerian equidistribution of Foata (1977) and a tempting bi-symmetry conjecture, which asserts that the quadruples $(asc,rep,zero,max)$ and $(rep,asc,max,zero)$ are equidistributed on ascent sequences.
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Recently, Yan and the first named author investigated systematically the enumeration of inversion or ascent sequences avoiding vincular patterns of length $3$, where two of the three letters are required to be adjacent. They established many connections with familiar combinatorial families and proposed several interesting conjectures. The objective of this paper is to address two of their conjectures concerning the enumeration of $underline{12}0$-avoiding inversion or ascent sequences.
In 1997, Bousquet-Melou and Eriksson stated a broad generalization of Eulers distinct-odd partition theorem, namely the $(k,l)$-Euler theorem. Their identity involved the $(k,l)$-lecture-hall partitions, which, unlike usual difference conditions of partitions in Rogers-Ramanujan type identities, satisfy some ratio constraints. In a 2008 paper, in response to a question suggested by Richard Stanley, Savage and Yee provided a simple bijection for the $l$-lecture-hall partitions (the case $k=l$), whose specialization in $l=2$ corresponds to Sylvesters bijection. Subsequently, as an open question, a generalization of their bijection was suggested for the case $k,lgeq 2$. In the spirit of Savage and Yees work, we provide and prove in this paper slight variations of the suggested bijection, not only for the case $k,lgeq 2$ but also for the cases $(k,1)$ and $(1,k)$ with $kgeq 4$. Furthermore, we show that our bijections equal the recursive bijections given by Bousquet-Melou and Eriksson in their recursive proof of the $(k,l)$-lecture hall and finally provide the analogous recursive bijection for the $(k,l)$-Euler theorem.
In 1997 Clarke et al. studied a $q$-analogue of Eulers difference table for $n!$ using a key bijection $Psi$ on symmetric groups. In this paper we extend their results to the wreath product of a cyclic group with the symmetric group. In particular we obtain a new mahonian statistic emph{fmaf} on wreath products. We also show that Foata and Hans two recent transformations on the symmetric groups provide indeed a factorization of $Psi$.
Permutation statistics $wnm$ and $rlm$ are both arising from permutation tableaux. $wnm$ was introduced by Chen and Zhou, which was proved equally distributed with the number of unrestricted rows of a permutation tableau. While $rlm$ is showed by Nadeau equally distributed with the number of $1$s in the first row of a permutation tableau. In this paper, we investigate the joint distribution of $wnm$ and $rlm$. Statistic $(rlm,wnm,rlmin,des,(underline{321}))$ is shown equally distributed with $(rlm,rlmin,wnm,des,(underline{321}))$ on $S_n$. Then the generating function of $(rlm,wnm)$ follows. An involution is constructed to explain the symmetric property of the generating function. Also, we study the triple statistic $(wnm,rlm,asc)$, which is shown to be equally distributed with $(rlmax-1,rlmin,asc)$ as studied by Josuat-Verg$grave{e}$s. The main method we adopt throughout the paper is constructing bijections based on a block decomposition of permutations.
We study statistics on ordered set partitions whose generating functions are related to $p,q$-Stirling numbers of the second kind. The main purpose of this paper is to provide bijective proofs of all the conjectures of stein (Arxiv:math.CO/0605670). Our basic idea is to encode ordered partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structure. We also give a partition version of MacMahons theorem on the equidistribution of the statistics inversion number and major index on words.
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