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A072590 Table T(n,k) giving number of spanning trees in complete bipartite graph K(n,k), read by antidiagonals.

%I A072590 #43 Feb 16 2025 08:32:46
%S A072590 1,1,1,1,4,1,1,12,12,1,1,32,81,32,1,1,80,432,432,80,1,1,192,2025,4096,
%T A072590 2025,192,1,1,448,8748,32000,32000,8748,448,1,1,1024,35721,221184,
%U A072590 390625,221184,35721,1024,1,1,2304,139968,1404928,4050000,4050000
%N A072590 Table T(n,k) giving number of spanning trees in complete bipartite graph K(n,k), read by antidiagonals.
%D A072590 J. W. Moon, Counting Labeled Trees, Issue 1 of Canadian mathematical monographs, Canadian Mathematical Congress, 1970.
%D A072590 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Exercise 5.66.
%H A072590 T. D. Noe, <a href="/A072590/b072590.txt">Antidiagonals d=1..50, flattened</a>
%H A072590 Taylor Brysiewicz and Aida Maraj, <a href="https://arxiv.org/abs/2310.13064">Lawrence Lifts, Matroids, and Maximum Likelihood Degrees</a>, arXiv:2310.13064 [math.CO], 2023. See p. 13.
%H A072590 H. I. Scoins, <a href="https://doi.org/10.1017/S0305004100036173">The number of trees with nodes of alternate parity</a>, Proc. Cambridge Philos. Soc. 58 (1962) 12-16.
%H A072590 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H A072590 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>
%F A072590 T(n, k) = n^(k-1) * k^(n-1).
%F A072590 E.g.f.: A(x,y) - 1, where: A(x,y) = exp( x*exp( y*A(x,y) ) ) = Sum_{n>=0} Sum_{k=0..n} (n-k)^k * (k+1)^(n-k-1) * x^(n-k)/(n-k)! * y^k/k!. - _Paul D. Hanna_, Jan 22 2019
%e A072590 From _Andrew Howroyd_, Oct 29 2019: (Start)
%e A072590 Array begins:
%e A072590 ============================================================
%e A072590 n\k | 1   2     3       4        5         6           7
%e A072590 ----+-------------------------------------------------------
%e A072590   1 | 1   1     1       1        1         1           1 ...
%e A072590   2 | 1   4    12      32       80       192         448 ...
%e A072590   3 | 1  12    81     432     2025      8748       35721 ...
%e A072590   4 | 1  32   432    4096    32000    221184     1404928 ...
%e A072590   5 | 1  80  2025   32000   390625   4050000    37515625 ...
%e A072590   6 | 1 192  8748  221184  4050000  60466176   784147392 ...
%e A072590   7 | 1 448 35721 1404928 37515625 784147392 13841287201 ...
%e A072590   ...
%e A072590 (End)
%t A072590 t[n_, k_] := n^(k-1) * k^(n-1); Table[ t[n-k+1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Feb 21 2013 *)
%o A072590 (PARI) {T(n, k) = if( n<1 || k<1, 0, n^(k-1) * k^(n-1))}
%Y A072590 Columns 2..3 are A001787, A069996.
%Y A072590 Main diagonal is A068087.
%Y A072590 Antidiagonal sums are A132609.
%Y A072590 Cf. A070285, A328887, A328888.
%K A072590 nonn,tabl,easy,nice
%O A072590 1,5
%A A072590 _Michael Somos_, Jun 23 2002
%E A072590 Scoins reference from _Philippe Deléham_, Dec 22 2003