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A210587 Triangle T(n,k) read by rows: T(n,k) is the number of unrooted hypertrees on n labeled vertices with k hyperedges, n >= 2, 1 <= k <= n-1.

%I A210587 #28 Feb 03 2025 09:48:11
%S A210587 1,1,3,1,12,16,1,35,150,125,1,90,900,2160,1296,1,217,4410,22295,36015,
%T A210587 16807,1,504,19264,179200,573440,688128,262144,1,1143,78246,1240029,
%U A210587 6889050,15707034,14880348,4782969,1,2550,302500,7770000,69510000,264600000,462000000,360000000,100000000
%N A210587 Triangle T(n,k) read by rows: T(n,k) is the number of unrooted hypertrees on n labeled vertices with k hyperedges, n >= 2, 1 <= k <= n-1.
%C A210587 See A210586 for the definition of a hypertree and for the enumeration of rooted hypertrees.
%H A210587 Andrew Howroyd, <a href="/A210587/b210587.txt">Table of n, a(n) for n = 2..1276</a>
%H A210587 Roland Bacher, <a href="https://arxiv.org/abs/1102.2708">On the enumeration of labelled hypertrees and of labelled bipartite trees</a>, arXiv:1102.2708 [math.CO], 2011.
%H A210587 J. McCammond and J. Meier, <a href="http://www.math.ucsb.edu/~mccammon/papers/htpandelltwo.pdf">The hypertree poset and the l^2-Betti numbers of the motion group of the trivial link</a>, Mathematische Annalen 328 (2004), no. 4, 633-652.
%F A210587 T(n,k) = n^(k-1)*Stirling2(n-1,k). T(n,k) = 1/n*A210586(n,k).
%F A210587 E.g.f.: A(x,t) = t + x*t^2/2! + (x + 3*x^2)*t^3/3! + ..., where t*d/dt(A(x,t)) is the e.g.f. for A210586.
%F A210587 Dobinski-type formula for the row polynomials: R(n,x) = exp(-n*x)*Sum_{k = 0..inf} n^(k-1)*k^(n-1)x^k/k!.
%F A210587 Row sums A030019.
%e A210587 Triangle begins
%e A210587 .n\k.|....1.....2......3......4......5......6
%e A210587 = = = = = = = = = = = = = = = = = = = = = = =
%e A210587 ..2..|....1
%e A210587 ..3..|....1.....3
%e A210587 ..4..|....1....12.....16
%e A210587 ..5..|....1....35....150....125
%e A210587 ..6..|....1....90....900...2160...1296
%e A210587 ..7..|....1...217...4410..22295..36015..16807
%e A210587 ...
%e A210587 Example of a hypertree with two hyperedges, one a 2-edge {3,4} and one a 3-edge {1,2,3}.
%e A210587 ........__________........................
%e A210587 ......./..........\.______................
%e A210587 ......|....1...../.\......\...............
%e A210587 ......|.........|.3.|....4.|..............
%e A210587 ......|....2.....\./______/...............
%e A210587 .......\__________/.......................
%e A210587 ..........................................
%e A210587 T(4,2) = 12. The twelve unrooted hypertrees on 4 vertices {1,2,3,4} with 2 hyperedges (one a 2-edge and one a 3-edge) have hyperedges:
%e A210587 {1,2,3} and {3,4}; {1,2,3} and {2,4}; {1,2,3} and {1,4};
%e A210587 {1,2,4} and {1,3}; {1,2,4} and {2,3}; {1,2,4} and {3,4};
%e A210587 {1,3,4} and {1,2}; {1,3,4} and {2,3}; {1,3,4} and {2,4};
%e A210587 {2,3,4} and {1,2}; {2,3,4} and {1,3}; {2,3,4} and {1,4}.
%p A210587 with(combinat):
%p A210587 A210587 := (n, k) -> n^(k-1)*stirling2(n-1, k):
%p A210587 for n from 2 to 10 do seq(A210587(n, k), k = 1..n-1) end do;
%p A210587 # _Peter Bala_, Oct 28 2015
%t A210587 T[n_, k_] := n^(k - 1)*StirlingS2[n - 1, k];
%t A210587 Table[T[n, k], {n, 2, 10}, {k, 1, n - 1}] // Flatten (* _Jean-François Alcover_, Sep 19 2019 *)
%o A210587 (PARI) T(n,k) = {n^(k-1)*stirling(n-1,k,2)}
%o A210587 for(n=2, 10, for(k=1, n-1, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Aug 28 2018
%Y A210587 Cf. A030019 (row sums). Cf. A210586, A048993.
%K A210587 nonn,easy,tabl
%O A210587 2,3
%A A210587 _Peter Bala_, Mar 26 2012