Pairings and Signed Permutations

Identity (1) is easy to prove: note that the left side is a convolution, so multiply it by xn , sum, and recognize the square of a binomial series. But a combinatorial interpretation of the same identity is not so easy to find. In [3], M. Sved recounts the story of the identity and its combinatorial proofs. She relates how, after she challenged her readers (in a previous article) to find a combinatorial proof, Paul Erdős “was quick to point out that . . . Hungarian mathematicians tackled it in the thirties: P. Veress proposing it, and G. Hajos solving it” [3, p. 44]. In the same article, she outlines more than one combinatorial proof supplied by her readers. Identity (1) has been mentioned by many other authors. We refer readers to [2, Exercise 2c, p. 44], [1] (in the foreword by D. Knuth) for recent citations. In this note, we describe a new combinatorial construction from which (1) is readily derived.