Jacobian elliptic functions and a family of bivariate peak polynomials
This paper has several combinatorial aims. After a long introduction with Taylor expansions of Jacobi elliptic functions, a combinatorial expansion for a family of bivariate peak polynomials, which count permutations by their odd and even cycle peaks, is given. This combinatorial expansion implies that the coefficient polynomials of sn\((u , k)\) are \(\gamma\)-positive. Furthermore, the coefficient polynomials of cn\((u , k)\) are bi-\(\gamma\)-positive. Then combinatorial interpretations of some numbers in terms of increasing trees are presented. Finally, combinatorial interpretations of two associated coefficients in terms of increasing trees are found. The proofs use the derivative operator acting on commuting variables, context-free grammar, grammatical labeling and induction.
scientific article; zbMATH DE number 7379248
scientific article