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Identities involving Narayana polynomials and Catalan numbers

2009
*
Discrete Mathematics
*

Then we represent

doi:10.1016/j.disc.2008.12.006
fatcat:armzyyrl5zhhtk4vqcn5xgyaxa
*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). ...##
###
Identities involving Narayana polynomials and Catalan numbers
[article]

2008
*
arXiv
*
pre-print

Then we represent

arXiv:0805.1274v1
fatcat:uswphgd4dnawtll3vgccniimce
*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. ...##
###
Restricted Permutations

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. ...

##
###
Combinatorial proofs of some Bell number formulas
[article]

2014
*
arXiv
*
pre-print

In this note, we provide bijective proofs of some

arXiv:1401.6588v1
fatcat:6aexbwgmnzcjjaxu6skeuoskg4
*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. ...##
###
A Combinatorial Proof for the Alternating Convolution of the Central Binomial Coefficients

2014
*
The American mathematical monthly
*

We give a combinatorial proof of

doi:10.4169/amer.math.monthly.121.06.537
fatcat:f3hzdhforjghrg7q4fq5gihmtm
*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] . ...##
###
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]

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*. ...

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

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. ...

##
###
The Delta Conjecture at q=1
[article]

2016
*
arXiv
*
pre-print

We use a weight-preserving,

arXiv:1609.04865v1
fatcat:6mvygghcuveovo3vi64zhk3zga
*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. ...##
###
Abacus-histories and the combinatorics of creation operators

2021
*
Journal of combinatorial theory. Series A
*

We develop

doi:10.1016/j.jcta.2020.105330
fatcat:25uyobstybeajfrtb26o37cbzq
*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. ...##
###
Enumeration and Special Functions
[chapter]

2003
*
Lecture notes in mathematics
*

Suppose that ϕ 1

doi:10.1007/3-540-44945-0_4
fatcat:potmo3iuu5aofjnqlznexuccmi
*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. ...##
###
Tableau sequences, open diagrams, and Baxter families

2016
*
European journal of combinatorics (Print)
*

Remarkably

doi:10.1016/j.ejc.2016.05.011
fatcat:7ewn6ibhizef7ctmnjldciqbeu
*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. ...##
###
Duplicial algebras and Lagrange inversion
[article]

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. ...

##
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An Extension of the Lindström-Gessel-Viennot Theorem

2022
*
Electronic Journal of Combinatorics
*

entries are

doi:10.37236/10913
fatcat:w7ufxfuc3fatnphurlhxs54ncu
*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. ...##
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Determinant Identities for Toeplitz-Hessenberg Matrices with Tribonacci Number Entries
[article]

2020
*
arXiv
*
pre-print

These determinant formulas may also be expressed equivalently as

arXiv:2003.10660v1
fatcat:rqhd33ocmvf6fiedgkjr3sk22q
*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. ...
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