A Shuffle Theorem for Paths Under Any Line (original) (raw)

A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

Canadian Journal of Mathematics, 2011

We introduce a q, t-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory ∇ operator applied to a Hall–Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the “shuffle conjecture” (Duke J. Math. 126 (2005), pp. 195 − 232) for ∇ en[X]. We bring to light that certain generalized Hall–Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of q, t-Catalan sequences, and we prove a number of identities involving these functions.

Bell polynomials and -generalized Dyck paths

Discrete Applied Mathematics, 2008

A k-generalized Dyck path of length n is a lattice path from (0, 0) to (n, 0) in the plane integer lattice Z × Z consisting of horizontal-steps (k, 0) for a given integer k ≥ 0, up-steps (1, 1), and down-steps (1, −1), which never passes below the x-axis. The present paper studies three kinds of statistics on k-generalized Dyck paths: "number of u-segments", "number of internal u-segments" and "number of (u, h)-segments". The Lagrange inversion formula is used to represent the generating function for the number of k-generalized Dyck paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. Many important special cases are considered leading to several surprising observations. Moreover, enumeration results related to u-segments and (u, h)-segments are also established, which produce many new combinatorial identities, and specially, two new expressions for Catalan numbers.

Dyck Paths with Forced and Forbidden Touch Points and q, t-Catalan building blocks

2010

We introduce a q, t-enumeration of Dyck paths which are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory ∇ operator applied to a Hall-Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the "shuffle conjecture" (Duke J. Math. 126 (2005), pp. 195 − 232) for ∇en[X]. We bring to light that certain generalized Hall-Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of q, t-Catalan sequences and we prove a number of identities involving these functions.

Ascending runs in permutations and valued Dyck paths

Ars Mathematica Contemporanea

We define a bijection between permutations and valued Dyck paths, namely, Dyck paths whose odd vertices are labelled with an integer that does not exceed their height. This map allows us to characterize the set of permutations avoiding the pattern 132 as the preimage of the set of Dyck paths with minimal labeling. Moreover, exploiting this bijection we associate to the set of n-permutations a polynomial that generalizes at the same time Eulerian polynomials, Motzkin numbers, super-Catalan numbers, little Schröder numbers, and other combinatorial sequences. Lastly, we determine the Hankel transform of the sequence of such polynomials.

Enumeration of Colored Dyck Paths Via Partial Bell Polynomials

Developments in Mathematics, 2019

We consider a class of lattice paths with certain restrictions on their ascents and down steps and use them as building blocks to construct various families of Dyck paths. We let every building block Pj take on cj colors and count all of the resulting colored Dyck paths of a given semilength. Our approach is to prove a recurrence relation of convolution type, which yields a representation in terms of partial Bell polynomials that simplifies the handling of different colorings. This allows us to recover multiple known formulas for Dyck paths and related lattice paths in an unified manner.

Refinements of -Dyck paths

European Journal of Combinatorics, 2011

The classical Chung-Feller theorem tells us that the number of (n, m)-Dyck paths is the nth Catalan number and independent of m. In this paper, we consider refinements of (n, m)-Dyck paths by using four parameters, namely the peak, valley, double descent and double ascent. Let p n,m,k be the total number of (n, m)-Dyck paths with k peaks. First, we derive the reciprocity theorem for the polynomial P n,m (x) = ∑ n k=1 p n,m,k x k. In particular, we prove that the number of (n, m)-Dyck paths with k peaks is equal to the number of (n, n − m)-Dyck paths with n − k peaks. Then we find the Chung-Feller properties for the sum of p n,m,k and p n,m,n−k , i.e., the number of (n, m)-Dyck paths which have k or n − k peaks is 2(n+2) n(n−1)  n k−1   n k+1  for 1 ≤ m ≤ n − 1 and independent of m. Finally, we provide a Chung-Feller type theorem for Dyck paths of semilength n with k double ascents: the total number of (n, m)-Dyck paths with k double ascents is equal to the total number of n-Dyck paths that have k double ascents and never pass below the x-axis, which is counted by the Narayana number. Let v n,m,k (resp. d n,m,k) be the total number of (n, m)-Dyck paths with k valleys (resp. double descents). Some similar results are derived.

On the combinatorics of modified lattice paths and generalized qqq--series

Contributions to Discrete Mathematics, 2018

Recently, Agarwal and Sachdeva, 2017, proved two Rogers-Ramanujan type identities for modified lattice paths by establishing a bijection between split (n + t)-color partitions and the modified lattice paths. In this paper, we interpret four generalized basic series combinatorially in terms of modified lattice paths by using a similar bijection. This leads to four new Rogers-Ramanujan type identities for modified lattice paths.

Exterior Pairs and Up Step Statistics on Dyck Paths

The Electronic Journal of Combinatorics, 2011

Let mathcalCn\mathcal{C}_nmathcalCn be the set of Dyck paths of length nnn. In this paper, by a new automorphism of ordered trees, we prove that the statistic 'number of exterior pairs', introduced by A. Denise and R. Simion, on the set mathcalCn\mathcal{C}_nmathcalCn is equidistributed with the statistic 'number of up steps at height hhh with hequiv0h\equiv 0hequiv0 (mod 3)'. Moreover, for mge3m\ge 3mge3, we prove that the two statistics 'number of up steps at height hhh with hequiv0h\equiv 0hequiv0 (mod mmm)' and 'number of up steps at height hhh with hequivm−1h\equiv m-1hequivm−1 (mod mmm)' on the set mathcalCn\mathcal{C}_nmathcalCn are 'almost equidistributed'. Both results are proved combinatorially.

An area-to-inv bijection between Dyck paths and 312-avoiding permutations

JOURNAL OF COMBINATORICS, 2001

The symmetric q, t-Catalan polynomial C n (q, t), which specializes to the Catalan polynomial C n (q) when t = 1, was defined by Garsia and Haiman in 1994. In 2000, Garsia and Haglund described statistics a(π) and b(π) on Dyck paths such that C n (q, t) = π q a(π) t b(π) where the sum is over all n × n Dyck paths. Specializing t = 1 gives the Catalan polynomial C n (q) defined by Carlitz and Riordan and further studied by Carlitz. Specializing both t = 1 and q = 1 gives the usual Catalan number C n . The Catalan number C n is known to count the number of n × n Dyck paths and the number of 312-avoiding permutations in S n , as well as at least 64 other combinatorial objects. In this paper, we define a bijection between Dyck paths and 312-avoiding permutations which takes the area statistic a(π) on Dyck paths to the inversion statistic on 312-avoiding permutations. The inversion statistic can be thought of as the number of (21) patterns in a permutation σ. We give a characterization for the number of (321), (4321), . . . , (k · · · 21) patterns that occur in σ in terms of the corresponding Dyck path.