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Plane trivalent trees and their patterns

Charles Delorme (2010)

Open Mathematics

The aim of this paper is to characterize the patterns of successive distances of leaves in plane trivalent trees, and give a very short characterization of their parity pattern. Besides, we count how many trees satisfy some given sequences of patterns.

Polynomial analogues of Ramanujan congruences for Han's hooklength formula

William J. Keith (2013)

Acta Arithmetica

This article considers the eta power ( 1 - q k ) b - 1 . It is proved that the coefficients of q n / n ! in this expression, as polynomials in b, exhibit equidistribution of the coefficients in the nonzero residue classes mod 5 when n = 5j+4. Other symmetries, as well as symmetries for other primes and prime powers, are proved, and some open questions are raised.

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