Lattice Paths and Negatively Indexed Weight-Dependent Binomial Coefficients

• Published in: Annals of combinatorics Vol. 27; no. 4; pp. 917 - 955
• Main Authors: Küstner, Josef, Schlosser, Michael J., Yoo, Meesue
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2023; Springer Nature B.V
• Subjects: Binomial coefficients; Combinatorial analysis; Combinatorics; Integers; Mathematical analysis; Mathematics; Mathematics and Statistics; symmetric functions; Binomial coefficients; elliptic binomial coefficient; 11B65; lliptic-commuting variables; stirling numbers; 33E05; Secondary 05E05; elliptic hypergeometric series; 11B73; Primary 05E16; commuting variables; commutation relations
• ISSN: 0218-0006; 0219-3094
• DOI: 10.1007/s00026-023-00639-1

In 1992, Loeb (Adv Math, 91:64–74, 1992) considered a natural extension of the binomial coefficients to negative entries and gave a combinatorial interpretation in terms of hybrid sets. He showed that many of the fundamental properties of binomial coefficients continue to hold in this extended setting. Recently, Formichella and Straub (Ann Comb, 23:725–748, 2019) showed that these results can be extended to the q -binomial coefficients with arbitrary integer values and extended the work of Loeb further by examining the arithmetic properties of the q -binomial coefficients. In this paper, we give an alternative combinatorial interpretation in terms of lattice paths and consider an extension of the more general weight-dependent binomial coefficients, first defined by Schlosser (Sém Lothar Combin, 81:24, 2020), to arbitrary integer values. Remarkably, many of the results of Loeb, Formichella and Straub continue to hold in the general weighted setting. We also examine important special cases of the weight-dependent binomial coefficients, including ordinary, q - and elliptic binomial coefficients as well as elementary and complete homogeneous symmetric functions.