Edges in the poset of partitions of an integer

The edges in the Hasse diagram of the partitions of an integer ordered by reverse refinement are studied. A recursive count of the edges between ranks is developed. Let P, be the poset of partitions of n, ordered by refinement. It contains p(n; n-t) elements of rank t, where p(n; k) denotes the number of partitions of n into h-parts, p(n) will denote the number of partitions of n. The number NE(n; t) of edges in the Hasse diagram of P, between rank levels t and t-1 is investigated [ 11. Considering the partitions of k into two parts, it is clear that p(k; 2) = Lk/2J An edge in the Hasse diagram corresponds to a two-part refinement of some candidate part. A particular candidate value k will appear in ~(n-k; n-t-1) partitions. Thus NE(b; t)= c p(k;2)p(n-kk;n-t-1). k>O Using p(n;m) =p(n-1; m-1) +p(n-mm; m) the following recursion results: NE(n;t)=NE(n-l;t)+NE(t+1;2t-n+l). Table I is an array of values of NE for some pairs of n; t. Maximally connected partitions in the Hasse diagram are complicated to describe in general. However, for 12 a triangle number there is a unique