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Advances in Applied Mathematics, ISSN 0196-8858, 06/2018, Volume 97, pp. 54 - 63
For integers with and , an -Dyck path is a lattice path in the integer lattice using up steps and down steps that goes from the origin to the point and... 
Lattice paths | Peaks | Chung–Feller theorem | [formula omitted]-Dyck paths | Bijective proof | (n,m)-Dyck paths | MATHEMATICS, APPLIED | Chung Feller theorem | (n, m)-Dyck paths
Journal Article
Discrete Applied Mathematics, ISSN 0166-218X, 08/2015, Volume 190-191, pp. 42 - 49
Recently Mansour and Shattuck studied -paths and gave formulas that related the total number of humps in all -paths to the number of super -paths. These... 
Motzkin paths | Peaks | [formula omitted]-Dyck paths | Humps | Narayana number | [formula omitted]-paths | (n, m) -Dyck paths | (k, a) -paths | MATHEMATICS, APPLIED | (n, m)-Dyck paths | COUNTING HUMPS | (k, a)-paths
Journal Article
Advances in Applied Mathematics, ISSN 0196-8858, 02/2020, Volume 113, p. 101975
Journal Article
Discrete Applied Mathematics, ISSN 0166-218X, 2008, Volume 156, Issue 12, pp. 2279 - 2292
A of length is a lattice path from to in the plane integer lattice consisting of horizontal-steps for a given integer , up-steps , and down-steps , which never... 
Catalan numbers | Potential polynomials | Bell polynomials | [formula omitted]-paths | k-paths | MOTZKIN | MATHEMATICS, APPLIED | bell polynomials | potential polynomials | NUMBERS | Mathematics - Combinatorics
Journal Article
Advances in Mathematics, ISSN 0001-8708, 10/2015, Volume 284, pp. 159 - 185
We define a family of maps on lattice paths, called , that assign levels to each step in the path and sort steps according to their level. Surprisingly,... 
Lattice paths | [formula omitted]-Catalan numbers | Diagonal harmonics | Sorting algorithms | Dyck paths | Q, t-Catalan numbers | MATHEMATICS | COMPACTIFIED JACOBIANS | STATISTICS | PATHS | PROOF | q, t-Catalan numbers | Q,T-CATALAN NUMBERS | CONJECTURE | Algorithms
Journal Article
Discrete Mathematics, ISSN 0012-365X, 11/2013, Volume 313, Issue 22, pp. 2552 - 2565
We prove a generalization of a conjecture of Dokos, Dwyer, Johnson, Sagan, and Selsor giving a recursion for the inversion polynomial of 321-avoiding... 
Generating function | Permutation | Inversion number | Catalan number | Major index | Motzkin path | Dyck path | Pattern avoidance | [formula omitted]-analogue | Continued fraction | Polyomino | q-analogue | DYCK PATHS | NUMBERS | PATTERNS | MATHEMATICS | CONTINUED FRACTIONS | RESTRICTED PERMUTATIONS | Permutations | Polynomials | Recursive | Recursion | Mathematical analysis | Inversions
Journal Article
Advances in Applied Mathematics, ISSN 0196-8858, 08/2015, Volume 69, pp. 65 - 108
Journal Article
Discrete Applied Mathematics, ISSN 0166-218X, 2007, Volume 155, Issue 17, pp. 2187 - 2201
We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition, which turns out to have many... 
Chain | Plane tree | Schröder path | [formula omitted]-Colored plane tree | Butterfly decomposition | Dyck path | Doubly rooted plane tree | k-Colored plane tree | plane tree | MATHEMATICS, APPLIED | chain | LATTICE PATHS | butterfly decomposition | NUMBERS | Schroder path | ENUMERATION | PARTITIONS | doubly rooted plane tree
Journal Article
Transactions of the American Mathematical Society, ISSN 0002-9947, 05/2019, Volume 372, Issue 5, pp. 3369 - 3404
Journal Article
Journal Article
Journal of Combinatorial Theory, Series A, ISSN 0097-3165, 05/2013, Volume 120, Issue 4, pp. 816 - 842
We classify recurrent configurations of the sandpile model on the complete bipartite graph in which one designated vertex is a sink. We present a bijection... 
Sandpile model | [formula omitted]-Narayana polynomial | Complete bipartite graph | Parallelogram polyomino | Recurrent states | Q, t-Narayana polynomial | BIJECTIONS | MATHEMATICS | MATRICES | q, t-Narayana polynomial | ENUMERATION | Combinatorics | Mathematics
Journal Article
Discrete Mathematics, ISSN 0012-365X, 07/2015, Volume 338, Issue 7, pp. 1197 - 1215
In this paper, we study the distribution of the number occurrences of the simplest frame pattern, called the pattern, in -cycles. Given an -cycle , we say that... 
[formula omitted]-analogue | Permutation pattern | Cycle | q-analogue | MATHEMATICS
Journal Article
Advances in Applied Mathematics, ISSN 0196-8858, 2010, Volume 44, Issue 1, pp. 16 - 36
The operator ∇ of F. Bergeron, Garsia, Haiman and Tesler [F. Bergeron, A. Garsia, M. Haiman, G. Tesler, Identities and positivity conjectures for some... 
Nabla operator | [formula omitted]-Catalan numbers | k-Schur functions | Level | Diagonal harmonics | (q, t)-Catalan numbers | MATHEMATICS, APPLIED | Statistics | Analysis | Universities and colleges | Mathematics - Combinatorics
Journal Article
Discrete Applied Mathematics, ISSN 0166-218X, 10/2015, Volume 193, pp. 208 - 212
Recently, Mansour and Shattuck related the total number of humps in all of the -paths of order to the number of super -paths, which generalized previous... 
Hump | Peak | [formula omitted]-path | (k, a) -path | MATHEMATICS, APPLIED | (k, a)-path | MOTZKIN PATHS
Journal Article
The Ramanujan Journal, ISSN 1382-4090, 12/2000, Volume 4, Issue 4, pp. 421 - 427
We study the combinatorics of a continued fraction formula due to Wall. We also derive the orthogonality of little q-Jacobi polynomials from this formula, as... 
Fourier Analysis | Functions of a Complex Variable | little q -Jacobi polynomials | Field Theory and Polynomials | Mathematics | Dyck path | Number Theory | Combinatorics | continued fraction formula | Continued fraction formula | Little q-Jacobi polynomials | MATHEMATICS | little q-Jacobi polynomials
Journal Article
Discrete Mathematics, ISSN 0012-365X, 2008, Volume 308, Issue 22, pp. 5218 - 5229
As a generalization of Haglund's statistic on Dyck paths [Conjectured statistics for the -Catalan numbers, Adv. Math. 175 (2) (2003) 319–334; A positivity... 
Involution | [formula omitted]-Analogue | Longest increasing subsequence | Forbidding pattern | Schröder | q, t-Analogue | Schroder | MATHEMATICS | longest increasing subsequence | NUMBERS | q,t-analogue | forbidding pattern | involution
Journal Article
Discrete Mathematics, ISSN 0012-365X, 2008, Volume 308, Issue 11, pp. 2222 - 2227
Riordan paths are Motzkin paths without horizontal steps on the -axis. We establish a correspondence between Riordan paths and -avoiding derangements. We also... 
Riordan number | (321,3 [formula omitted]42)-avoiding derangement | Riordan path | (321,3over(1, -)42)-avoiding derangement | MOTZKIN | MATHEMATICS | TREES | NUMBERS | (321,3over-bar42)-avoiding derangement | RECURRENCES
Journal Article
Discrete Mathematics, ISSN 0012-365X, 2008, Volume 308, Issue 24, pp. 5954 - 5964
Many interesting combinatorial objects are enumerated by the -Catalan numbers, one possible generalization of the Catalan numbers. We will present a new... 
[formula omitted]-ary tree | [formula omitted]-ary number | Staircase tiling | Set-valued Young tableau | [formula omitted]-Catalan number | k-Catalan number | k-ary number | k-ary tree | MATHEMATICS | WEYL GROUPS | NUMBERS | ENUMERATION | T-ARY TREES | RANKING
Journal Article
Discrete Mathematics, ISSN 0012-365X, 03/2020, Volume 343, Issue 3, p. 111728
In this work, we generalize and utilize the linear relations of LLT polynomials introduced by Lee (2017). By using the fact that the chromatic quasisymmetric... 
LLT polynomials | [formula omitted]-positivity | Lollipop graph | Natural unit interval order | Chromatic quasisymmetric function | [formula omitted]-free poset
Journal Article
Discrete Mathematics, ISSN 0012-365X, 03/2020, Volume 343, Issue 3, p. 111731
We present a new general approach for the enumeration of directed convex polyominoes, which lets us easily control several statistics. This method relies on a... 
[formula omitted]-convex polyominoes | Generating functions | Fibonacci polynomials
Journal Article
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