346 research outputs found
Descent sets on 321-avoiding involutions and hook decompositions of partitions
We show that the distribution of the major index over the set of involutions
in S_n that avoid the pattern 321 is given by the q-analogue of the n-th
central binomial coefficient. The proof consists of a composition of three
non-trivial bijections, one being the Robinson-Schensted correspondence,
ultimately mapping those involutions with major index m into partitions of m
whose Young diagram fits inside an n/2 by n/2 box. We also obtain a refinement
that keeps track of the descent set, and we deduce an analogous result for the
comajor index of 123-avoiding involutions
Enumeration of Standard Young Tableaux
A survey paper, to appear as a chapter in a forthcoming Handbook on
Enumeration.Comment: 65 pages, small correction
Euler-Mahonian Statistics On Ordered Set Partitions (II)
We study statistics on ordered set partitions whose generating functions are
related to -Stirling numbers of the second kind. The main purpose of this
paper is to provide bijective proofs of all the conjectures of \stein
(Arxiv:math.CO/0605670). Our basic idea is to encode ordered partitions by a
kind of path diagrams and explore the rich combinatorial properties of the
latter structure. We also give a partition version of MacMahon's theorem on the
equidistribution of the statistics inversion number and major index on words.Comment: 27 pages,8 figure
The excedances and descents of bi-increasing permutations
Starting from some considerations we make about the relations between certain
difference statistics and the classical permutation statistics we study
permutations whose inversion number and excedance difference coincide. It turns
out that these (so-called bi-increasing) permutations are just the 321-avoiding
ones. The paper investigates their excedance and descent structure. In
particular, we find some nice combinatorial interpretations for the
distribution coefficients of the number of excedances and descents,
respectively, and their difference analogues over the bi-increasing
permutations in terms of parallelogram polyominoes and 2-Motzkin paths. This
yields a connection between restricted permutations, parallelogram polyominoes,
and lattice paths that reveals the relations between several well-known
bijections given for these objects (e.g. by Delest-Viennot,
Billey-Jockusch-Stanley, Francon-Viennot, and Foata-Zeilberger). As an
application, we enumerate skew diagrams according to their rank and give a
simple combinatorial proof for a result concerning the symmetry of the joint
distribution of the number of excedances and inversions, respectively, over the
symmetric group.Comment: 36 page
Three dimensional Narayana and Schröder numbers
AbstractConsider the 3-dimensional lattice paths running from (0,0,0) to (n,n,n), constrained to the region {(x,y,z):0⩽x⩽y⩽z}, and using various step sets. With C(3,n) denoting the set of constrained paths using the steps X≔(1,0,0), Y≔(0,1,0), and Z≔(0,0,1), we consider the statistic counting descents on a path P=p1p2…p3n∈C(3,n), i.e., des(P)≔|{i:pipi+1∈{YX,ZX,ZY},1⩽i⩽3n-1}|. A combinatorial cancellation argument and a result of MacMahon yield a formula for the 3-Narayana number, N(3,n,k)≔|{P∈C(3,n):des(P)=k+2}|. We define other statistics distributed by the 3-Narayana number and show that 4∑k2kN(3,n,k) yields the nth large 3-Schröder number which counts the constrained paths using the seven positive steps of the form (ξ1,ξ2,ξ3), ξi∈{0,1}
Proof of a conjectured q,t-Schr\"{o}der identity
A conjecture of Chunwei Song on a limiting case of the q,t-Schr\"{o}der
theorem is proved combinatorially. The proof matches pairs of tableaux to
Catalan words in a manner that preserves differences in the maj statistic.Comment: 8 pages; v2 corrects presentation error in example and notation
(substance of proof unchanged
- …