Catalan triangle numbers and binomial coefficients

Kyu Hwan Lee, Se Jin Oh

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

4 Scopus citations

Abstract

The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac-Moody algebras. We prove that any binomial coefficient can be written as weighted sums along rows of the Catalan triangle. The coefficients in the sums form a triangular array, which we call the alternating Jacobsthal triangle. We study various subsequences of the entries of the alternating Jacobsthal triangle and show that they arise in a variety of combinatorial constructions. The generating functions of these sequences enable us to define their k-analogue of q-deformation. We show that this deformation also gives rise to interesting combinatorial sequences. The starting point of this work is certain identities in the study of Khovanov-Lauda-Rouquier algebras and fully commutative elements of a Coxeter group.

Original languageEnglish
Title of host publicationContemporary Mathematics
PublisherAmerican Mathematical Society
Pages165-185
Number of pages21
DOIs
StatePublished - 2018

Publication series

NameContemporary Mathematics
Volume713
ISSN (Print)0271-4132
ISSN (Electronic)1098-3627

Bibliographical note

Funding Information:
The first author was partially supported by a grant from the Simons Foundation (#318706). The second author was supported by NRF Grant # 2016R1C1B2013135.

Publisher Copyright:
© 2018 American Mathematical Society.

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