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Báo cáo toán học: "New directions in enumerative chess problems to Richard Stanley on the occasion of his 60th birthday"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: New directions in enumerative chess problems to Richard Stanley on the occasion of his 60th birthday. | New directions in enumerative chess problems to Richard Stanley on the occasion of his 60th birthday Noam D. Elkies Department of Mathematics Harvard University Cambridge MA 02138 elkies@math.harvard.edu Submitted Jun 30 2005 Accepted Aug 1 2005 Published Aug 24 2005 Mathematics Subject Classifications 05A10 05A15 05E10 97A20 Abstract Normally a chess problem must have a unique solution and is deemed unsound even if there are alternatives that differ only in the order in which the same moves are played. In an enumerative chess problem the set of moves in the solution is usually unique but the order is not and the task is to count the feasible permutations via an isomorphic problem in enumerative combinatorics. Almost all enumerative chess problems have been series-movers in which one side plays an uninterrupted series of moves unanswered except possibly for one move by the opponent at the end. This can be convenient for setting up enumeration problems but we show that other problem genres also lend themselves to composing enumerative problems. Some of the resulting enumerations cannot be shown or have not yet been shown in series-movers. This article is based on a presentation given at the banquet in honor of Richard Stanley s 60th birthday and is dedicated to Stanley on this occasion. 1 Motivation and overview Normally a chess problem must have a unique solution and is deemed unsound even if there are alternatives that differ only in the order in which the same moves are played. In an enumerative chess problem the set of moves in the solution is usually unique but the order is not and the task is to count the feasible permutations via an isomorphic problem in enumerative combinatorics. Quite a few such problems have been composed and published since about 1980 see for instance Puu St4 . As Stanley notes in St4 almost all such problems have been of a special type known as series-movers . In this article we give examples showing how several other kinds of problems .