A210586 - OEIS

%I #27 Oct 26 2024 04:46:05

%S 2,3,9,4,48,64,5,175,750,625,6,540,5400,12960,7776,7,1519,30870,

%T 156065,252105,117649,8,4032,154112,1433600,4587520,5505024,2097152,9,

%U 10287,704214,11160261,62001450,141363306,133923132,43046721,10,25500,3025000,77700000,695100000,2646000000,4620000000,3600000000,1000000000

%N Triangle T(n,k) read by rows: T(n,k) is the number of rooted hypertrees on n labeled vertices with k hyperedges, n >= 2, k >= 1.

%C A hypergraph H is a pair (V,E) consisting of a finite set V of vertices and a set E of hyperedges given by subsets of V containing at least two elements. A walk in a hypergraph H connecting vertices v0 and vn is a sequence v0, e1, v1, e2, ... , v(n-1), en, vn, where each vi is in V and each ei is in E and for each ei the set {v(i-1),vi} is contained in ei. If for every pair of vertices v and v0 there is a walk in H starting at v and ending at v0 then H is called connected. A walk is a cycle if it contains at least two edges, all of the ei are distinct and all of the vi are distinct except v0 = vn. A connected hypergraph with no cycles is called a hypertree. A rooted hypertree is a hypertree in which one particular vertex is selected as being the root. For the enumeration of unrooted hypertrees see A210587.

%H Andrew Howroyd, <a href="/A210586/b210586.txt">Table of n, a(n) for n = 2..1276</a>

%H R. Bacher, <a href="http://arxiv.org/abs/1102.2708">On the enumeration of labelled hypertrees and of labelled bipartite trees</a>, arXiv:1102.2708 [math.CO], 2011.

%H 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 T(n,k) = n^k*Stirling2(n-1,k). T(n,k) = n*A210587(n,k).

%F E.g.f. A(x,t) = t + 2*x*t^2/2! + (3*x + 9*x^2)*t^3/3! + ... satisfies A(x,t) = t*exp(x*(exp(A(x,t)) - 1)).

%F Dobinski-type formula for the row polynomials: R(n,x) = exp(-n*x)*Sum_{k = 0..inf} n^k*k^(n-1)x^k/k!.

%F Row sums A035051.

%F The e.g.f. is essentially the series reversion of t/F(x,t) w.r.t. t, where F(x,t) = exp(x*(exp(t) - 1)) is the e.g.f. of the Stirling numbers of the second kind A048993. - _Peter Bala_, Oct 28 2015

%e Triangle begins

%e .n\k.|....1.....2......3.......4.......5.......6

%e = = = = = = = = = = = = = = = = = = = = = = = = =

%e ..2..|....2

%e ..3..|....3.....9

%e ..4..|....4....48.....64

%e ..5..|....5...175....750.....625

%e ..6..|....6...540...5400...12960....7776

%e ..7..|....7..1519..30870..156065..252105..117649

%e ...

%e Example of a hypertree with two hyperedges, one a 2-edge {3,4} and one a 3-edge {1,2,3}.

%e ........__________........................

%e ......./..........\.______................

%e ......|....1...../.\......\...............

%e ......|.........|.3.|....4.|..............

%e ......|....2.....\./______/...............

%e .......\__________/.......................

%e ..........................................

%e T(4,2) = 48. 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 {1,2,3} and {3,4}; {1,2,3} and {2,4}; {1,2,3} and {1,4};

%e {1,2,4} and {1,3}; {1,2,4} and {2,3}; {1,2,4} and {3,4};

%e {1,3,4} and {1,2}; {1,3,4} and {2,3}; {1,3,4} and {2,4};

%e {2,3,4} and {1,2}; {2,3,4} and {1,3}; {2,3,4} and {1,4}.

%e Choosing one of the four vertices as the root gives a total of 4x12 = 48 rooted hypertrees on 4 vertices.

%p with(combinat):

%p A210586 := (n, k) -> n^k*stirling2(n-1, k):

%p for n from 2 to 10 do seq(A210586(n, k), k = 1..n-1) end do;

%p # _Peter Bala_, Oct 28 2015

%o (PARI) T(n,k) = {n^k*stirling(n-1,k,2)}

%o for(n=2, 10, for(k=1, n-1, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Aug 28 2018

%Y Cf. A035051 (row sums). Cf. A210587, A048993.

%K nonn,easy,tabl,changed

%O 2,1

%A _Peter Bala_, Mar 26 2012