Counting Even and Odd Partitions
نویسنده
چکیده
1. Introduction. It is a lovely fact that n] = f1; 2; : : : ; ng, n 1, has as many subsets X with an even cardinality jXj as those with an odd cardinality, namely 2 n?1 of both. To prove it, pair every subset X with X 1 where X 1 is Xnf1g if 1 2 X and X f1g if 1 6 2 X. Then X 7 ! X 1 is an involution that changes the parity of jXj and the result follows. More generally, in enumerative combinatorics one often has a family S n of objects on n] such that every object X has a natural size s(X) 2 N 0 of some kind. Then besides the total number of objects S n = jS n j one can
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ورودعنوان ژورنال:
- The American Mathematical Monthly
دوره 110 شماره
صفحات -
تاریخ انتشار 2003