The Eulerian numbers count the number of permutations in the symmetric groups with a certain number of descents. The generating function for the Eulerian numbers can be sorted into terms that correspond to equivalence classes partitioned by the particularly elegant “valley-hopping” proof. The two-sided Eulerian numbers are an analog of the Eulerian numbers that also count the number descents of each permutation’s inverse. Gessel’s conjecture sorts the generating function for the two-sided Eulerian numbers, and it has been proven using recurrence relations, but a proof that matches the valley-hopping proof in elegance is yet to be found.
Wagner, Haven, "Two-Sided Eulerian Numbers" (2017). Senior Independent Study Theses. Paper 7488.
Discrete Mathematics and Combinatorics
Bachelor of Arts
Senior Independent Study Thesis
© Copyright 2017 Haven Wagner