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Identities involving Narayana polynomials and Catalan numbers

Toufik Mansour, Yidong Sun
2009 Discrete Mathematics  
Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities.  ...  We give three different proofs for these identities, namely, two algebraic proofs and one combinatorial proof. Some applications are also given which lead to many known and new identities. Crown  ...  Acknowledgements The authors are grateful to the anonymous referees for the helpful suggestions and comments. The second author was supported by The National Science Foundation of China (10801020).  ... 
doi:10.1016/j.disc.2008.12.006 fatcat:armzyyrl5zhhtk4vqcn5xgyaxa

Identities involving Narayana polynomials and Catalan numbers [article]

Toufik Mansour, Yidong Sun
2008 arXiv   pre-print
Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities.  ...  We give three different proofs for these identities, namely, two algebraic proofs and one combinatorial proof. Some applications are also given which lead to many known and new identities.  ...  Acknowledgements The second author was supported by The National Science Foundation of China.  ... 
arXiv:0805.1274v1 fatcat:uswphgd4dnawtll3vgccniimce

Restricted Permutations

Rodica Simion, Frank W. Schmidt
1985 European journal of combinatorics (Print)  
The number of even and odd such permutations is found and the involutions among them are counted. Bijections are determined between sets of such permutations and other combinatorial objects.  ...  The results in this paper complement previous work by Knuth, Lovasz, Rotem and Stanley.  ...  Two such bijections will involve the 0-1 strings counted by the Fibonacci numbers, and the subsets of a (n -1 )-element set.  ... 
doi:10.1016/s0195-6698(85)80052-4 fatcat:o2yioct5dnfy3lx5hqno4axgku

Combinatorial proofs of some Bell number formulas [article]

Mark Shattuck
2014 arXiv   pre-print
In this note, we provide bijective proofs of some identities involving the Bell number, as previously requested.  ...  We also provide a further interpretation for a related difference of Catalan numbers in terms of the inclusion-exclusion principle.  ...  In [6] , combinatorial proofs were sought for some identities involving Bell numbers and binomial coefficients.  ... 
arXiv:1401.6588v1 fatcat:6aexbwgmnzcjjaxu6skeuoskg4

A Combinatorial Proof for the Alternating Convolution of the Central Binomial Coefficients

Michael Z. Spivey
2014 The American mathematical monthly  
We give a combinatorial proof of the identity for the alternating convolution of the central binomial coefficients.  ...  Our proof entails applying an involution to certain colored permutations and showing that only permutations containing cycles of even length remain.  ...  The author originally posted this proof on the mathematics question-and-answer site, Mathematics Stack Exchange [4] .  ... 
doi:10.4169/amer.math.monthly.121.06.537 fatcat:f3hzdhforjghrg7q4fq5gihmtm

Page 41 of Mathematical Reviews Vol. , Issue 92a [page]

1992 Mathematical Reviews  
In fact, using a sign-reversing involution on a certain set which involves integer partitions, the authors provide a bijective proof of an alternating sign version of the Pfaff-Saalschiitz 392-summation  ...  One application is counting the number as- signed to each of many categories of a very large number of events.  ... 

Moments of Askey-Wilson polynomials [article]

Jang Soo Kim, Dennis Stanton
2013 arXiv   pre-print
New formulas for the nth moment of the Askey-Wilson polynomials are given.  ...  These are derived using analytic techniques, and by considering three combinatorial models for the moments: Motzkin paths, matchings, and staircase tableaux.  ...  Kim's sign-reversing involution. We now recall his involution.  ... 
arXiv:1207.3446v2 fatcat:4qe6xnlozngfzexsfg3ze5piza

The mysterious story of square ice, piles of cubes, and bijections

Ilse Fischer, Matjaž Konvalinka
2020 Proceedings of the National Academy of Sciences of the United States of America  
two-dimensional ice) and their relations to certain classes of plane partitions.  ...  When combinatorialists discover two different types of objects that are counted by the same numbers, they usually want to prove this by constructing an explicit bijective correspondence.  ...  We are also grateful to the referees for reading the manuscript so carefully and for a number of suggested improvements.  ... 
doi:10.1073/pnas.2005525117 pmid:32879007 fatcat:htmzmvq6pjehngnnbg3sioxw4m

The Delta Conjecture at q=1 [article]

Marino Romero
2016 arXiv   pre-print
We use a weight-preserving, sign-reversing involution to find a combinatorial expansion of Δ_e_k e_n at q=1 in terms of the elementary symmetric function basis.  ...  The method of proof provides a variety of structures which can compute the inner product of Δ_e_k e_n|_q=1 with any symmetric function.  ...  The sign-reversing involution is used to cancel all the negative signs and output a finite number of fixed points yielding a polynomial in t with positive integer coefficients.  ... 
arXiv:1609.04865v1 fatcat:6mvygghcuveovo3vi64zhk3zga

Abacus-histories and the combinatorics of creation operators

Nicholas A. Loehr, Gregory S. Warrington
2021 Journal of combinatorial theory. Series A  
We develop new combinatorial models for the Schur expansions of these and related symmetric functions using objects called abacus-histories.  ...  We use involutions on abacus-histories to give bijective proofs of properties of the Bernstein creation operator and Hall-Littlewood polynomials indexed by three-row partitions.  ...  C b and B b , that play a crucial role in the study of q, t-Catalan numbers, diagonal harmonics modules, and the Bergeron-Garsia nabla operator.  ... 
doi:10.1016/j.jcta.2020.105330 fatcat:25uyobstybeajfrtb26o37cbzq

Enumeration and Special Functions [chapter]

Dennis Stanton
2003 Lecture notes in mathematics  
Suppose that ϕ 1 and ϕ 2 are two sign-reversing involutions on A, with fixed point sets F P (ϕ 1 ) ⊂ A + and F P (ϕ 2 ) ⊂ A + respectively.  ...  A sign-reversing involution ϕ is a map ϕ : A → A such that ϕ 2 = ϕ and if ϕ(x) = x then sign(x)sign(ϕ(x)) = −1. This just means that ϕ changes sign on its orbits of size 2.  ... 
doi:10.1007/3-540-44945-0_4 fatcat:potmo3iuu5aofjnqlznexuccmi

Tableau sequences, open diagrams, and Baxter families

Sophie Burrill, Julien Courtiel, Eric Fusy, Stephen Melczer, Marni Mishna
2016 European journal of combinatorics (Print)  
Remarkably two subclasses of open diagrams are equinumerous with well known objects: standard Young tableaux of bounded height, and Baxter permutations.  ...  We show that walks that start at ∅, end at a row shape, and only visit partitions of bounded height are in bijection with a new type of arc diagram -- open diagrams.  ...  JC is supported by the ANR GRAAl, ANR-14-CE25-0014-02, and by the PIMS postdoctoral fellowship grant.  ... 
doi:10.1016/j.ejc.2016.05.011 fatcat:7ewn6ibhizef7ctmnjldciqbeu

Duplicial algebras and Lagrange inversion [article]

Jean-Christophe Novelli, Jean-Yves Thibon
2020 arXiv   pre-print
The Catalan subalgebra is identified with the free duplicial algebra on one generator, and the Schr\"oder subalgebra is interpreted by means of a new operad, which we call triduplicial.  ...  We provide operadic interpretations for two Hopf subalgebras of the algebra of parking functions.  ...  and Catalan numbers respectively.  ... 
arXiv:1209.5959v3 fatcat:2pmd2b27ezb7vdle2r62v2jote

An Extension of the Lindström-Gessel-Viennot Theorem

Yi-Lin Lee
2022 Electronic Journal of Combinatorics  
entries are signed counts of lattice paths with given starting and ending points.  ...  We give a formula for the total weight of the families of non-intersecting paths on $G$ with any given starting and ending points.  ...  The author also thanks the reviewers for carefully reading the manuscript and providing helpful comments.  ... 
doi:10.37236/10913 fatcat:w7ufxfuc3fatnphurlhxs54ncu

Determinant Identities for Toeplitz-Hessenberg Matrices with Tribonacci Number Entries [article]

Taras Goy, Mark Shattuck
2020 arXiv   pre-print
These determinant formulas may also be expressed equivalently as identities that involve sums of products of multinomial coefficients and tribonacci numbers.  ...  In particular, we establish a connection between the tribonacci and the Fibonacci and Padovan sequences via Toeplitz-Hessenberg determinants.  ...  If r = 5 and n ≥ 4, then U contains two elements of opposite sign, if n is even, and of the same sign, if n is odd.  ... 
arXiv:2003.10660v1 fatcat:rqhd33ocmvf6fiedgkjr3sk22q
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