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

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

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Enumerations of humps and peaks in (k,a)-paths and (n,m)-Dyck paths via bijective proofs

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

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

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

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

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

In this article we introduce the -cover poset of an arbitrary bounded poset , which is a certain subposet of the -fold direct product of with itself. Its...

Fuß–Catalan combinatorics | m-Tamari lattice | EL-shellability | Path poset | Left-modularity | Northeast paths | Möbius function | Trimness | Chord posets | m-cover poset | Rooted Trees | m-Dyck paths | [formula omitted]-free posets | posets | FuCatalan combinatorics | (2 +2)-free posets | Chord | Mbius function | MATHEMATICS, APPLIED | (2+2)-free posets | COMPLEXES | ELEMENTS | Fu beta-Catalan combinatorics | Mobius function | Mathematics - Combinatorics

Fuß–Catalan combinatorics | m-Tamari lattice | EL-shellability | Path poset | Left-modularity | Northeast paths | Möbius function | Trimness | Chord posets | m-cover poset | Rooted Trees | m-Dyck paths | [formula omitted]-free posets | posets | FuCatalan combinatorics | (2 +2)-free posets | Chord | Mbius function | MATHEMATICS, APPLIED | (2+2)-free posets | COMPLEXES | ELEMENTS | Fu beta-Catalan combinatorics | Mobius function | Mathematics - Combinatorics

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

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

We introduce new families of combinatorial objects whose enumeration computes volumes of flow polytopes. These objects provide an interpretation, based on...

unified diagram | binomial transform | Chan-Robbins-Yuen polytope | PARKING FUNCTIONS | Kostant partition function | Lidskii formula | log-concave | multi-labeled Dyck path | parking function | gravity diagram | Tesler polytope | Flow polytope | Ehrhart polynomial | parking triangle | MATHEMATICS | Pitman-Stanley polytope | zigzag graph | line-dot diagram | Dyck path | Catalan numbers | caracol graph

unified diagram | binomial transform | Chan-Robbins-Yuen polytope | PARKING FUNCTIONS | Kostant partition function | Lidskii formula | log-concave | multi-labeled Dyck path | parking function | gravity diagram | Tesler polytope | Flow polytope | Ehrhart polynomial | parking triangle | MATHEMATICS | Pitman-Stanley polytope | zigzag graph | line-dot diagram | Dyck path | Catalan numbers | caracol graph

Journal Article

Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), ISSN 1815-0659, 03/2016, Volume 12

We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin...

Cauchy’s type kernels and symmetric | Schubert | NilCoxeter and idCoxeter algebras | Di FrancescoZinn-Justin polynomials | Alternating sign matrices | Double affine nilCoxeter algebras | Noncrossing Dyck paths and (rectangular) Schubert polynomials | Plactic monoid and reduced plactic algebras | β-Grothendieck | Totally symmetric plane partitions | Key and (double) key-Grothendieck | SYMMETRY CLASSES | and alternating sign matrices | SCHUR-FUNCTIONS | key and (double) key-Grothendieck | Cauchy's type kernels and symmetric | multi-parameter deformations of Genocchi numbers of the first and the second types | ALGEBRA | VARIETIES | FORMULA | PHYSICS, MATHEMATICAL | CHARACTERS | double affine nilCoxeter algebras | totally symmetric plane partitions | RING | plactic monoid and reduced plactic algebras | ALTERNATING-SIGN MATRICES | YOUNG TABLEAUX | nilCoxeter and idCoxeter algebras | beta-Grothendieck | Gandhi-Dumont polynomials and (staircase) Schubert polynomials | noncrossing Dyck paths and (rectangular) Schubert polynomials | OPERATORS | and Di Francesco-Zinn-Justin polynomials | Mathematics - Combinatorics

Cauchy’s type kernels and symmetric | Schubert | NilCoxeter and idCoxeter algebras | Di FrancescoZinn-Justin polynomials | Alternating sign matrices | Double affine nilCoxeter algebras | Noncrossing Dyck paths and (rectangular) Schubert polynomials | Plactic monoid and reduced plactic algebras | β-Grothendieck | Totally symmetric plane partitions | Key and (double) key-Grothendieck | SYMMETRY CLASSES | and alternating sign matrices | SCHUR-FUNCTIONS | key and (double) key-Grothendieck | Cauchy's type kernels and symmetric | multi-parameter deformations of Genocchi numbers of the first and the second types | ALGEBRA | VARIETIES | FORMULA | PHYSICS, MATHEMATICAL | CHARACTERS | double affine nilCoxeter algebras | totally symmetric plane partitions | RING | plactic monoid and reduced plactic algebras | ALTERNATING-SIGN MATRICES | YOUNG TABLEAUX | nilCoxeter and idCoxeter algebras | beta-Grothendieck | Gandhi-Dumont polynomials and (staircase) Schubert polynomials | noncrossing Dyck paths and (rectangular) Schubert polynomials | OPERATORS | and Di Francesco-Zinn-Justin polynomials | Mathematics - Combinatorics

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

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

[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

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

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

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

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

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

[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

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

[formula omitted]-convex polyominoes | Generating functions | Fibonacci polynomials

Journal Article

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