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Some permutation problems
Authors:L Carlitz  Richard Scoville
Institution:Department of Mathematics, Duke University, Durham, North Carolina 27706 USA
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), a > π(a), or a = π(a). Let A(r, s, t) be the number of πZn 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 πZn 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.
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