Abstract: 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:
(Ⅰ)H_n,m=H_1,m=∑_i=1^m|△i|_m, m ≥ n;0,m<n;H_1,m=(3·2^m-1(2m)!)/(m!(m+2)!),m ≥ 1,for n ≥ 2,where
|△i|_m=((2i-2)!)/((i-1)|i|){(4i-2)α(i+1,m-i)+α(i,m-i)-iα(i,m-i+1)+(i+1)β(i,m-i)},m>i,Meanwhile, the combinatorial identity
(Ⅱ)(3·2ⁿ(2m-1)/(m+2)+4m-1-m²)((2m-2)!)/((m-1)!(m+1)!)+∑_i=2^m-1|△i|_m=((2m)!)/(m!(m+1)!) is also found. In what mentioned above, α(s, t,) and β(s, t) are expressed by the following finite sums with all the terms positive:
α(s,t)=∑_j=0^t-1((2t)!(2t+s-j-1)!)/(t(t-j-1)|j|(2t-j)|(t+s)|);β(s,t)=∑_j=0^t-1((2t+1)!(2t+s-j))/(|t(t-j-1)|j|(2t-j+1)|(t+s+1)!).