Related papers: On the number of even roots of permutations

We study generating functions for the number of even (odd) permutations on n letters avoiding 132 and an arbitrary permutation $\tau$ on k letters, or containing $\tau$ exactly once. In several interesting cases the generating function…

We study the generating function for the number of even (or odd) permutations on n letters containing exactly $r\gs0$ occurrences of 132. It is shown that finding this function for a given r amounts to a routine check of all permutations in…

The number of $k$-roots of an arbitrary permutation is expressed as an alternating sum of $\mu$-unimodal $k$-roots of the identity permutation.

Let m be a fixed positive integer. It is well-known that a permutation $\sigma$ may have one, many, or no mth roots. In this note we provide an explicit expression and a generating function for the number of mth roots of \sigma. Let p_m(n)…

Let $S_n$ denote the group all permutations of $n$. For every permutation $\sigma$, we let $\mathrm{des}(\sigma)$ denote the number of descents in $\sigma$ and $\mathrm{LRMin}(\sigma)$ denote the number of left-to-right minima of $\sigma$.…

We study generating functions for the number of permutations on n letters avoiding 132 and an arbitrary permutation $\tau$ on k letters, or containing $\tau$ exactly once. In several interesting cases the generating function depends only on…

We give a syntactic view of the Sawada-Williams $(\sigma,\tau)$-generation of permutations. The corresponding sequence of $\sigma-\tau$-operations, of length $n!-1$ is shown to be highly compressible: it has $O(n^2\log n)$ bit description.…

We present general links between statistics of non-Hermitian random matrices and the distribution of the number of cycles of some specific random permutations. In particular, we derive explicit formulas for the generating functions of the…

Let R(n,k) be the number of permutations of $\{1,2,\ldots,n\}$ with k alternating runs. In this paper, we establish the relationships between R(n,k) and the central factorial numbers of even indices as well as the number of signed…

We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. The generating function can…

In this paper, we find an explicit formulas, or recurrences, in terms of generating functions for the cardinalities of the sets $S_n(T;\tau)$ of all permutations in $S_n$ that contain $\tau\in S_k$ exactly once and avoid a subset…

We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation sigma = sigma_1sigma_2...sigma_n defined as the set of indices…

We consider two related problems arising from a question of R. Graham on quasirandom phenomena in permutation patterns. A ``pattern'' in a permutation $\sigma$ is the order type of the restriction of $\sigma : [n] \to [n]$ to a subset $S…

A permutation \tau contains another permutation \sigma as a pattern if \tau has a subsequence whose elements are in the same order with respect to size as the elements in \sigma. This defines a partial order on the set of all permutations,…

For each integer k >= 2, let F(k) denote the largest n for which there exists a permutation \sigma \in S_n, all of whose patterns of length k are distinct. We prove that F(k) = k + \lfloor \sqrt{2k-3} \rfloor + e_k, where e_k \in {-1,0} for…

Let \tau(.) be the Ramanujan \tau-function, and let k be a positive integer such that \tau(n) is not 0 for n=1,...,[k/2]. (This is known to be true for k < 10^{23}, and, conjecturally, for all k.) Further, let s be a permutation of the set…

A permutation sigma in Sn is a k-derangement if for any subset X = {a1, . . ., ak} \subseteq [n], {sigma(a1), . . ., sigma(ak)} is not equal to X. One can form the k-derangement graph on the set of permutations of Sn by connecting two…

Let T_k^m={\sigma \in S_k | \sigma_1=m}. We prove that the number of permutations which avoid all patterns in T_k^m equals (k-2)!(k-1)^{n+1-k} for k <= n. We then prove that for any \tau in T_k^1 (or any \tau in T_k^k), the number of…

Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We study generating functions for the number…

A simple permutation is one which maps no proper non-singleton interval onto an interval. We consider the enumeration of simple permutations from several aspects. Our results include a straightforward relationship between the ordinary…