70 research outputs found
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 of a graph
and asked whether there are non-isomorphic trees and with . 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
-polynomials, and prove three main theorems. First, for all non-empty
connected graphs , Stanley's polynomial is irreducible
in for all large enough . 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
, , and for ) 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
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