Acta mathematica scientia,Series B ›› 1992, Vol. 12 ›› Issue (1): 85-88.

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A NOTE ON THE NUMBER OF BIPARTITE PLANAR MAPS

Liu Yanpei   

  1. Inst. of Appl. Math., Academia Sinica, Beijing 100080, China
  • Received:1998-01-24 Online:1992-03-25 Published:1992-03-25
  • Supported by:
    Project supported by the NNSF,China.

Abstract: Let Hn,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)Hn,m=H1,m=∑i=1m|i|m, mn;0,m<n;H1,m=(3·2m-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=2m-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=0t-1((2t)!(2t+s-j-1)!)/(t(t-j-1)|j|(2t-j)|(t+s)|);β(s,t)=∑j=0t-1((2t+1)!(2t+s-j))/(|t(t-j-1)|j|(2t-j+1)|(t+s+1)!).

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