Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions

Abstract

We find Stieltjes-type and Jacobi-type continued fractions for some "master polynomials" that enumerate permutations, set partitions or perfect matchings with a large (sometimes infinite) number of simultaneous statistics. Our results contain many previously obtained identities as special cases, providing a common refinement of all of them.

arXiv commentLaTeX2e, 122 pages, includes 9 tikz figures. Version 2 includes a new footnote 27 on page 81. To be published in Advances in Applied Mathematics

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• Lattice paths and branched continued fractions. III. Generalizations of the Laguerre, rook and Lah polynomials
• Linear extensions and continued fractions
• A remark on continued fractions for permutations and D-permutations with a weight $-1$ per cycle
• Continued fractions for cycle-alternating permutations
• Continued fractions using a Laguerre digraph interpretation of the Foata--Zeilberger bijection and its variants
• Equidistributions of mesh patterns of length two and Kitaev and Zhang's conjectures
• Equidistributions around special kinds of descents and excedances
• Bijective Enumeration and Sign-Imbalance for Permutation Depth and Excedances