Richard Scoville

Enumeration of pairs of sequences by rises, falls and levels

The paper is concerned with the enumeration of pairs of sequences with given specification according to rises, falls and levels. Thus there are nine possibilities RR, ..., LL. Generating functions in the general case are very complicated. However in a number of special cases simple explicit results are obtained.

Generating functions for certain types of permutations

Generating functions are obtained for certain types of permutations analogous to up-down and down-up permutations. In each case the generating function is a quotient of entire functions; the denominator in each case is φ02(x) − φ1(x)φ3(x), where φj(x)=∑n=o∞x4n+j(4n+j)!.

Some permutation problems

Abstract Put Zn = {1, 2,…, n} and let π denote an arbitrary permutation of Zn. Problem I. Let π = (π(1), π(2), …, π(n)). π has an up, down, or fixed point at a according as a π(a), or a = π(a). Let A (r, s, t) be the number of π ∈ Zn with r ups, s downs, and t fixed points. Problem II. Consider the triple π−1(a), a, π(a). Let R denote an up and F a down of π and let B(n, r, s) denote the number of π ∈ Zn with r occurrences of π−1(a)RaRπ(a) and s occurrences of π−1(a)FaFπ(a). Generating functions are obtained for each enumerant as well as for a refinement of the second. In each case use is made of the cycle structure of permutations.