@inbook{c5a4608aa3b64f4f9d18a6afbde949d8,

title = "Catalan triangle numbers and binomial coefficients",

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.",

author = "Lee, {Kyu Hwan} and Oh, {Se Jin}",

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: {\textcopyright} 2018 American Mathematical Society.",

year = "2018",

doi = "10.1090/conm/713/14315",

language = "English",

series = "Contemporary Mathematics",

publisher = "American Mathematical Society",

pages = "165--185",

booktitle = "Contemporary Mathematics",

}