A Note on Stephan’s Conjecture 87

Recently Stephan [5] posted 117 conjectures based on an extensive analysis of the On-line Encyclopedia of Integer Sequences [3, 4]. Here we prove conjecture 87. Let Kn,m denote a complete bipartite graph with part sizes n and m, and let Pn denote a path with n vertices. Fix an integer k > 1. Here we are concerned with counting perfect matchings on the graphs Gn = K1,k−1 × P2n. (For k > 2, there are no perfect matchings on K1,k−1 × P2n+1, which is bipartite with parts of unequal size.) For k = 2 and k = 3, this problem is equivalent to the extremely well-studied problems of counting domino tilings of a 2-by-2n or 3-by-2n grid, respectively. See [2], section 7.1, for an extensive discussion. We give two versions of the central combinatorial argument (Lemma 1 and the first proof of Lemma 3). The second argument, whose form was suggested by Henry Cohn, is simpler. However, the work of the first yields some information on the structure of the matchings (Corollary 2). The two recurrences derived are easily seen to be equivalent (second proof of Lemma 3), so we follow only one path to the generating function (Proposition 4). Let’s name the vertices of Gn. Call c1, c2, . . . , c2n the centers, and call di,j , 1 ≤ i ≤ 2n, 1 ≤ j ≤ k − 1 be the peripheral vertices. There two types of edges: • Horizontal: {ci, di,j}, for i = 1, . . . , 2n and j = 1, . . . , k − 1. • Vertical: {ci, ci+1} and {di,j , di+1,j} for 1 ≤ i ≤ 2n − 1, 1 ≤ j ≤ k − 1. We say {ci, ci+1} and {di,j , di+1,j} are at level i.) Lemma 1. Let k > 1 be a positive integer and let an be the number of perfect matchings of the graph Gn. Then a0 = 1, a1 = k,