Balanced Properties of the q-Derangement Numbers and the q-Catalan Numbers
William Y.C. Chen, David G.L. Wang, and Larry X.W. Wang
TL;DR
This paper establishes a general criterion for the balanced property of permutation statistics and demonstrates that q-derangement and q-Catalan numbers satisfy this property, contributing to combinatorial enumeration theory.
Contribution
It introduces a new criterion for the balanced property based on generating functions and applies it to q-derangement and q-Catalan numbers.
Findings
q-derangement numbers satisfy the balanced property
q-Catalan numbers satisfy the balanced property
Provides a general criterion for balanced properties in permutation statistics
Abstract
Based on Bona's condition for the balanced property of the number of cycles of permutations, we give a general criterion for the balanced property in terms of the generating function of a statistic. We show that the q-derangement numbers and the q-Catalan numbers satisfy the balanced property.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Identities · semigroups and automata theory