A symmetry on weakly increasing trees and multiset Schett polynomials

By considering the parity of the degrees and levels of nodes in increasing trees, a new combinatorial interpretation for the coefficients of the Taylor expansions of the Jacobi elliptic functions is found. As one application of this new interpretation, a conjecture of Ma–Mansour–Wang–Yeh is solved. Unifying the concepts of increasing trees and plane trees, Lin–Ma–Ma–Zhou introduced weakly increasing trees on a multiset. A symmetry joint distribution of “even-degree nodes on odd levels” and “odd-degree nodes” on weakly increasing trees is found, extending the Jacobi elliptic functions to multisets. A combinatorial proof and an algebraic proof of this symmetry are provided, as well as several relevant interesting consequences.

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