Tiling problems, automata, and tiling graphs

This paper continues the investigation of tiling problems via formal languages, which was begun in papers by Merlini, Sprugnoli, and Verri. Those authors showed that certain tiling problems could be encoded by regular languages, which lead automatically to generating functions and other combinatorial information on tilings. We introduce a method of simplifying the DFA's recognizing these language, which leads to bijective proofs of certain tiling identities. We apply these ideas to some other tiling problems, including three-dimensional tilings and tilings with triangles and rhombi. We also study graph-theoretic variations of these tiling problems

A Bijective Proof of a Factorization Formula for Specialized Macdonald Polynomials

Let μ and ν = (ν 1, . . . , ν k ) be partitions such that μ is obtained from ν by adding m parts of sizer. Descouens and Morita proved algebraically that the modified Macdonald polynomials H~μ(X;q,t) satisfy the identity H~μ=H~νH~(rm) when the parameter t is specialize to an mth root of unity. Descouens, Morita, and Numata proved this formula bijectively when r ≤ ν k and r∈{1,2}. This note gives a bijective proof of the formula for all r ≤ ν k

New combinatorial formulations of the shuffle conjecture

The shuffle conjecture (due to Haglund, Haiman, Loehr, Remmel, and Ulyanov) provides a combinatorial formula for the Frobenius series of the diagonal harmonics module DHn, which is the symmetric function∇(en). This formula is a sum over all labeled Dyck paths of terms built from combinatorial statistics called area, dinv, and IDes. We provide three new combinatorial formulations of the shuffle conjecture based on other statistics on labeled paths, parking functions, and related objects. Each such reformulation arises by introducing an appropriate new definition of the inverse descent set. Analogous results are proved for the higher-order shuffle conjecture involving ∇m(en). We also give new versions of some recently proposed combinatorial formulas for ∇(Cα) and ∇(s(k,1(n−k))), which translate expansions based on the dinv statistic into equivalent expansions based on Haglund\u27s bounce statistic

A continuous family of partition statistics equidistributed with length

AbstractThis article investigates a remarkable generalization of the generating function that enumerates partitions by area and number of parts. This generating function is given by the infinite product ∏i⩾11/(1−tqi). We give uncountably many new combinatorial interpretations of this infinite product involving partition statistics that arose originally in the context of Hilbert schemes. We construct explicit bijections proving that all of these statistics are equidistributed with the length statistic on partitions of n. Our bijections employ various combinatorial constructions involving cylindrical lattice paths, Eulerian tours on directed multigraphs, and oriented trees