摘要
Let H(n,m) be the number of rooted non-isomorphic bipartite planar maps with m edges and the valency of the rooted face being 2n. This note provides the following results: [GRAPHICS] for n greater-than-or-equal-to 2, where \DELTA(i)\m = (2i - 2)! divided-by (i - 1)!i! {(4i - 2)alpha(i + 1, m - i) + alpha(i, m - i) - i-alpha(i, m - i + 1) + (i + 1)beta(i, m - i)}, m > i, (2i - 2)! divided-by (i - 1) !i! (4i - 1 - i2), m = i;0, m < i. [GRAPHICS] Meanwhile, the combinatorial identity [GRAPHICS] is also found. In what mentioned above, alpha(s, t) and beta(s, t) are expressed by the following finite sums with all the terms positive:(~)[GRAPHICS]
Let H(n,m) be the number of rooted non-isomorphic bipartite planar maps with m edges and the valency of the rooted face being 2n. This note provides the following results: [GRAPHICS] for n greater-than-or-equal-to 2, where \DELTA(i)\m = (2i - 2)! divided-by (i - 1)!i! {(4i - 2)alpha(i + 1, m - i) + alpha(i, m - i) - i-alpha(i, m - i + 1) + (i + 1)beta(i, m - i)}, m > i, (2i - 2)! divided-by (i - 1) !i! (4i - 1 - i2), m = i;0, m < i. [GRAPHICS] Meanwhile, the combinatorial identity [GRAPHICS] is also found. In what mentioned above, alpha(s, t) and beta(s, t) are expressed by the following finite sums with all the terms positive:(~)[GRAPHICS]
