23 11 Article 05.1.5 Journal of Integer Sequences, Vol. 8 (2005), 2 3 6 1 47 Bijective Proofs of Parity Theorems for Partition Statistics Mark Shattuck Department of Mathematics University of Tennessee Knoxville, TN 37996-1300 USA shattuck@math.utk.edu Abstract We give bijective proofs of parity theorems for four related statistics on partitions of finite sets. A consequence of our results is a combinatorial proof of a congruence between Stirling numbers and binomial coefficients. 1 Introduction The notational conventions of this paper are as follows: N := {0, 1, 2,... }, P := {1, 2,... }, [0] := ∅, and [n] := {1,...,n} for n ∈ P. Empty sums take the value 0 and empty products the value 1, with 0 0 := 1. The binomial coefficient ( n k ) is equal to zero if k is a negative integer or if 0  n<k. Let Π(n, k) denote the set of all partitions of [n] with k blocks and Π(n) the set of all partitions of [n]. Associate to each π ∈ Π(n, k) the ordered partition (E 1 ,...,E k ) of [n] comprising the same blocks as π, arranged in increasing order of their smallest elements, and define statistics ˜ w,ˆ w, w ∗ , and w by ˜ w(π) := k  i=1 (i − 1)(|E i |− 1), (1.1) ˆ w(π) := k  i=1 i(|E i |− 1) = ˜ w(π)+ n − k, (1.2) w ∗ (π) := k  i=1 i|E i | =˜ w(π)+ n +  k 2  , (1.3) 1