Making simple proofs simpler

An open partition p [Cod09a, Cod09b] of a tree T is a partition of the vertices of T with the property that, for each block B of p, the upset of B is a union of blocks of p. This paper deals with the number, NP(n), of open partitions of the tree, Vn, made of two chains with n points each, that share the root. We prove that the number of such partitions is given by the formula NPn = k=1 n  j=1 n n 1 k 1 n 1 j 1 Mink, j Below are a few values of NP(n): 1, 5, 26, 130, 628, 2954, 13612, 61716, ... On the other hand, from the On-Line Encyclopedia of Integer Sequences [OEIS] we learn that such a numerical sequence is generated by the formula n + 1 22 n-3 n 1 2 2 n 2 n 1 In this note we exploit Mathematica to explain the not so simple derivation of the following novel identity, that will prove the equivalence between the above formulas: NPn = k=1 n  j=1 n n1 k 1 n 1 j 1 Mink, j = n + 1 22 n-3 - k=1 n  h=1 k-1 k h n1 h1 n 1 k 1 In so doing we show that visual enumeration can make simple proofs simpler!