## 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 language | English |
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Title of host publication | Contemporary Mathematics |

Publisher | American Mathematical Society |

Pages | 165-185 |

Number of pages | 21 |

DOIs | |

State | Published - 2018 |

### Publication series

Name | Contemporary Mathematics |
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Volume | 713 |

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.