Equidistribution of Descents, Adjacent Pairs, and Place-Value Pairs on Permutations
نویسندگان
چکیده
An (X, Y )-descent in a permutation is a pair of adjacent elements such that the first element is from X, the second element is from Y , and the first element is greater than the second one. An (X, Y )-adjacency in a permutation is a pair of adjacent elements such that the first one is from X and the second one is from Y . An (X, Y )place-value pair in a permutation is an element y in position x, such that y is in Y and x is in X. It turns out, that for certain choices of X and Y some of the three statistics above become equidistributed. Moreover, it is easy to derive the distribution formula for (X, Y )-place-value pairs thus providing distribution for other statistics under consideration too. This generalizes some results in the literature. As a result of our considerations, we get combinatorial proofs of several remarkable identities. We also conjecture existence of a bijection between two objects in question preserving a certain statistic.
منابع مشابه
Counting subwords in flattened permutations
In this paper, we consider the number of occurrences of descents, ascents, 123-subwords, 321-subwords, peaks and valleys in flattened permutations, which were recently introduced by Callan in his study of finite set partitions. For descents and ascents, we make use of the kernel method and obtain an explicit formula (in terms of Eulerian polynomials) for the distribution on Sn in the flattened ...
متن کاملPattern and position based permutation statistics
Falling permutation patterns of length two where we allow to mark adjacency both in position and value give rise to three statistics, namely the adjacency (πi − πi+1 = 1), the descent and the inversion. The two former are equidistributed with the position based statistics fixed points and excedances, and the latter is Mahonian and thus equidistributed with many statistics. We prove that the bis...
متن کاملPermutations Statistics of Indexed and Poset Permutations by Einar Steingrimsson
The definitions of descents and excedances in the elements of the symmetric group Sd are generalized in two different directions. First, descents and excedances are defined for indexed permutations, i.e. the elements of the group Sd = Zn I Sd, where is wreath product with respect to the usual action of Sd by permutation of [d]. It is shown, bijectively, that excedances and descents are equidist...
متن کاملVariations on Descents and Inversions in Permutations
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 σ = σ1σ2 · · · σn defined as the set of indices i such that either i is odd and σi > σi+1, or i is even and σi < σi+1. We show that this statistic is equidistributed with the odd 3-factor set statistic on permutations σ̃ ...
متن کاملNormal Approximations for Descents and Inversions of Permutations of Multisets
Normal approximations for descents and inversions of permutations of the set {1, 2, . . . , n} are well known. We consider the number of inversions of a permutation π(1), π(2), . . . , π(n) of a multiset with n elements, which is the number of pairs (i, j) with 1 ≤ i < j ≤ n and π(i) > π(j). The number of descents is the number of i in the range 1 ≤ i < n such that π(i) > π(i + 1). We prove tha...
متن کامل