On the joint distributions of succession and Eulerian statistics

The motivation of this paper is to investigate the joint distribution of succession and Eulerian statistics. We first investigate the enumerators for the joint distribution of descents, big ascents and successions over all permutations in the symmetric group. As an generalization a result of Diaconis-Evans-Graham (Adv. in Appl. Math., 61 (2014), 102-124), we show that two triple set-valued statistics of permutations are equidistributed on symmetric groups. We then introduce the definition of proper left-to-right minimum, and discover that the joint distribution of the succession and proper left-to-right minimum statistics over permutations is a symmetric distribution. In the final part, we discuss the relationship between the fix and cyc (p,q)-Eulerian polynomials and the joint distribution of succession and Eulerian-type statistics. In particular, we give a concise derivation of the generating function for a six-variable Eulerian polynomials.Comment: 21 pages. arXiv admin note: text overlap with arXiv:2002.0693

UNSEPARATED PAIRS AND FIXED POINTS IN RANDOM PERMUTATIONS

In a uniform random permutation Π of [n]: = {1, 2,..., n}, the set of elements k ∈ [n−1] such that Π(k +1) = Π(k)+1 has the same distribution as the set of fixed points of Π that lie in [n−1]. We give three different proofs of this fact using, respectively, an enumeration relying on the inclusion-exclusion principle, the introduction of two different Markov chains to generate uniform random permutations, and the construction of a combinatorial bijection. We also obtain the distribution of the analogous set for circular permutations that consists of those k ∈ [n] such that Π(k + 1 mod n) = Π(k) + 1 mod n. This latter random set is just the set of fixed points of the commutator [ρ, Π], where ρ is the n-cycle (1, 2,..., n). We show for a general permutation η that, under weak conditions on the number of fixed points and 2-cycles of η, the total variation distance between the distribution of the number of fixed points of [η, Π] and a Poisson distribution with expected value 1 is small when n is large