On Conjugates for Set Partitions and Integer Compositions

There is a familiar conjugate for integer partitions: transpose the Ferrers diagram, and a conjugate for integer compositions: transpose a Ferrers-like diagram. Here we propose a conjugate for set partitions and exhibit analogous pairs of statistics interchanged by the conjugate on set partitions and integer compositions respectively. 0 The Conjugate of an Integer Partition A partition of n is a weakly decreasing list of positive integers, called its parts, whose sum is n. The Ferrers diagram of a partition a1 a2 : : : ak 1 is the k-row left-justi ed array of dots with ai dots in the i-th row. The conjugate, obtained by transposing the Ferrers diagram, is a well known involution on partitions of n that interchanges the largest part and the number of parts. 1 A Conjugate for Set Partitions The partitions of an nelement set, say [n] = f1; 2; : : : ; ng, into nonempty blocks are counted by the Bell numbers, A000110 in OEIS. A singleton is a block containing just 1 element and an adjacency is an occurrence of two consecutive elements of [n] in the same block. Consecutive is used here in the cyclic sense so that n and 1 are also considered to be consecutive (the ordinary sense is considered in [2]). We say i initiates an adjacency if i and i+1 mod n are in the same block and analogously for terminating an adjacency. The number of k-block partitions of [n] containing no adjacencies is considered in a recent Monthly problem proposed