A BUDGET OF RHYME SCHEME COUNTS1

Le résumé fourni par la source

Summary Rhyme schemes, perhaps the oldest combinatorial setting behind the Setting and Bell numbers, are rescued from current neglect by this budget of enumerations. A rhyme scheme for a stanza of n verses is a (number) sequence (r1, r2, …, rn) in which rj may be any of 1, 2, …, dj‐1+ 1, with dj the number of distinct numbers among r1, …, rj, the rhyme schemes for n= 3 are: 111, 112, 121, 122, 123. The total number of schemes for n verses is of course the Bell number Bn. They have a trivial mapping of set‐partitions but some enumerations familiar in the study of sequences are odd (even idiotic) when rephrased as set‐partitions. The budget displayed (by no means exhaustive) is selected for its interest in the study of Bell and Stirling numbers. One surprise is the appearance of refined Bell multivariable polynomials.

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