Sweep maps: A continuous family of sorting algorithms

We define a family of maps on lattice paths, called sweep maps, that assign levels to each step in the path and sort steps according to their level. Surprisingly, although sweep maps act by sorting, they appear to be bijective in general. The sweep maps give concise combinatorial formulas for the q,t-Catalan numbers, the higher q,t-Catalan numbers, the q,t-square numbers, and many more general polynomials connected to the nabla operator and rational Catalan combinatorics. We prove that many algorithms that have appeared in the literature (including maps studied by Andrews, Egge, Gorsky, Haglund, Hanusa, Jones, Killpatrick, Krattenthaler, Kremer, Orsina, Mazin, Papi, Vaille, and the present authors) are all special cases of the sweep maps or their inverses. The sweep maps provide a very simple unifying framework for understanding all of these algorithms. We explain how inversion of the sweep map (which is an open problem in general) can be solved in known special cases by finding a "bounce path" for the lattice paths under consideration. We also define a generalized sweep map acting on words over arbitrary alphabets with arbitrary weights, which is also conjectured to be bijective.Comment: 21 pages; full version of FPSAC 2014 extended abstrac

Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials

We introduce explicit combinatorial interpretations for the coefficients in some of the transition matrices relating to skew Hall-Littlewood polynomials P_lambda/mu(x;t) and Hivert's quasisymmetric Hall-Littlewood polynomials G_gamma(x;t). More specifically, we provide: 1) the G-expansions of the Hall-Littlewood polynomials P_lambda, the monomial quasisymmetric polynomials M_alpha, the quasisymmetric Schur polynomials S_alpha, and the peak quasisymmetric functions K_alpha; 2) an expansion of P_lambda/mu in terms of the F_alpha's. The F-expansion of P_lambda/mu is facilitated by introducing starred tableaux.Comment: 28 pages; added brief discussion of the Hall-Littlewood Q', typos corrected, added references in response to referee suggestion

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

A rooted variant of Stanley's chromatic symmetric function

Richard Stanley defined the chromatic symmetric function $X_G$ of a graph $G$ and asked whether there are non-isomorphic trees $T$ and $U$ with $X_T=X_U$. We study variants of the chromatic symmetric function for rooted graphs, where we require the root vertex to either use or avoid a specified color. We present combinatorial identities and recursions satisfied by these rooted chromatic polynomials, explain their relation to pointed chromatic functions and rooted $U$-polynomials, and prove three main theorems. First, for all non-empty connected graphs $G$, Stanley's polynomial $X_G(x_1,\ldots,x_N)$ is irreducible in $\mathbb{Q}[x_1,\ldots,x_N]$ for all large enough $N$. The same result holds for our rooted variant where the root node must avoid a specified color. We prove irreducibility by a novel combinatorial application of Eisenstein's Criterion. Second, we prove the rooted version of Stanley's Conjecture: two rooted trees are isomorphic as rooted graphs if and only if their rooted chromatic polynomials are equal. In fact, we prove that a one-variable specialization of the rooted chromatic polynomial (obtained by setting $x_0=x_1=q$, $x_2=x_3=1$, and $x_n=0$ for $n>3$) already distinguishes rooted trees. Third, we answer a question of Pawlowski by providing a combinatorial interpretation of the monomial expansion of pointed chromatic functions.Comment: 21 pages; v2: added a short algebraic proof to Theorem 2 (now Theorem 15), we also answer a question of Pawlowski about monomial expansions; v3: added additional one-variable specialization results, simplified main proof