Adjacent q-cycles in permutations

TL;DR

This paper introduces a new permutation statistic counting q-cycles of consecutive integers, providing explicit formulas, recurrences, and generating functions, with generalizations to multiple fixed lengths.

Contribution

It defines a novel permutation statistic and derives comprehensive formulas and generating functions, extending to multiple cycle lengths.

Findings

Explicit formulas for the distribution of the new statistic

Recurrence relations and generating functions derived

Generalization to multiple fixed cycle lengths

Abstract

We introduce a new permutation statistic, namely, the number of cycles of length $q$ consisting of consecutive integers, and consider the distribution of this statistic among the permutations of $\{1,2,...,n\}$. We determine explicit formulas, recurrence relations, and ordinary and exponential generating functions. A generalization to more than one fixed length is also considered.