18 research outputs found
Symmetry properties of the Novelli-Pak-Stoyanovskii algorithm
The number of standard Young tableaux of a fixed shape is famously given by
the hook-length formula due to Frame, Robinson and Thrall. A bijective proof of
Novelli, Pak and Stoyanovskii relies on a sorting algorithm akin to
jeu-de-taquin which transforms an arbitrary filling of a partition into a
standard Young tableau by exchanging adjacent entries. Recently, Krattenthaler
and M\"uller defined the complexity of this algorithm as the average number of
performed exchanges, and Neumann and the author proved it fulfils some nice
symmetry properties. In this paper we recall and extend the previous results
and provide new bijective proofs.Comment: 13 pages, 3 figure, submitted to FPSAC 2014 Chicag
Rational Shi tableaux and the skew length statistic
International audienceWe define two refinements of the skew length statistic on simultaneous core partitions. The first one relies on hook lengths and is used to prove a refined version of the theorem stating that the skew length is invariant under conjugation of the core. The second one is equivalent to a generalisation of Shi tableaux to the rational level of Catalan combinatorics. We prove that the rational Shi tableau is injective. Moreover we present a uniform definition of the rational Shi tableau for Weyl groups and conjecture injectivity in the general case
-partitions and -positivity
Using the combinatorics of -unimodal sets, we establish two new
results in the theory of quasisymmetric functions. First, we obtain the
expansion of the fundamental basis into quasisymmetric power sums. Secondly, we
prove that generating functions of reverse -partitions expand positively
into quasisymmetric power sums. Consequently any nonnegative linear combination
of such functions is -positive whenever it is symmetric. As an application
we derive positivity results for chromatic quasisymmetric functions,
unicellular and vertical strip LLT polynomials, multivariate Tutte polynomials
and the more general -polynomials, matroid quasisymmetric functions, and
certain Eulerian quasisymmetric functions, thus reproving and improving on
numerous results in the literature.Comment: 47 pages, 4 figure
A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions
We give a new characterization of the vertical-strip LLT polynomials
as the unique family of symmetric functions that satisfy
certain combinatorial relations. This characterization is then used to prove an
explicit combinatorial expansion of vertical-strip LLT polynomials in terms of
elementary symmetric functions. Such formulas were conjectured independently by
A. Garsia et al. and the first named author, and are governed by the
combinatorics of orientations of unit-interval graphs. The obtained expansion
is manifestly positive if is replaced by , thus recovering a recent
result of M. D'Adderio. Our results are based on linear relations among LLT
polynomials that arise in the work of D'Adderio, and of E. Carlsson and A.
Mellit. To some extent these relations are given new bijective proofs using
colorings of unit-interval graphs. As a bonus we obtain a new characterization
of chromatic quasisymmetric functions of unit-interval graphs.Comment: 49 pages. This version has updated .bib, and some improvements in
section
Type C parking functions and a zeta map
We introduce type C parking functions, encoded as vertically labelled lattice
paths and endowed with a statistic dinv'. We define a bijection from type C
parking functions to regions of the Shi arrangement of type C, encoded as
diagonally labelled ballot paths and endowed with a natural statistic area'.
This bijection is a natural analogue of the zeta map of Haglund and Loehr and
maps dinv' to area'. We give three different descriptions of it.Comment: 12 page