An involution on Dyck paths and its consequences

AbstractAn involution is introduced in the set of all Dyck paths of semilength n from which one re-obtains easily the equidistribution of the parameters ‘number of valleys’ and ‘number of doublerises’ and also the equidistribution of the parameters ‘height of the first peak’ and ‘number of returns’

On divisibility of Narayana numbers by primes

Using Kummer's Theorem, we give a necessary and sufficient condition for a Narayana number to be divisible by a given prime. We use this to derive certain properties of the Narayana triangle.Comment: 5 pages, see related papers at http://www.math.msu.edu/~saga

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}

Hook formulas for skew shapes

International audienceThe celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give two q-analogues of Naruse's formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations

Motzkin Intervals and Valid Hook Configurations

We define a new natural partial order on Motzkin paths that serves as an intermediate step between two previously-studied partial orders. We provide a bijection between valid hook configurations of $312$-avoiding permutations and intervals in these new posets. We also show that valid hook configurations of permutations avoiding $132$ (or equivalently, $231$) are counted by the same numbers that count intervals in the Motzkin-Tamari posets that Fang recently introduced, and we give an asymptotic formula for these numbers. We then proceed to enumerate valid hook configurations of permutations avoiding other collections of patterns. We also provide enumerative conjectures, one of which links valid hook configurations of $312$-avoiding permutations, intervals in the new posets we have defined, and certain closed lattice walks with small steps that are confined to a quarter plane.Comment: 22 pages, 8 figure