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 A210586 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. 3
 2, 3, 9, 4, 48, 64, 5, 175, 750, 625, 6, 540, 5400, 12960, 7776, 7, 1519, 30870, 156065, 252105, 117649, 8, 4032, 154112, 1433600, 4587520, 5505024, 2097152, 9, 10287, 704214, 11160261, 62001450, 141363306, 133923132, 43046721, 10, 25500, 3025000, 77700000, 695100000, 2646000000, 4620000000, 3600000000, 1000000000 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS 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. LINKS R. Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011. J. McCammond and J. Meier, The hypertree poset and the l^2-Betti numbers of the motion group of the trivial link, Mathematische Annalen 328 (2004), no. 4, 633-652. FORMULA T(n,k) = n^k*Stirling2(n-1,k). T(n,k) = n*A210587(n,k). 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)). 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!. Row sums A035051. 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 EXAMPLE Triangle begins .n\k.|....1.....2......3.......4.......5.......6 = = = = = = = = = = = = = = = = = = = = = = = = = ..2..|....2 ..3..|....3.....9 ..4..|....4....48.....64 ..5..|....5...175....750.....625 ..6..|....6...540...5400...12960....7776 ..7..|....7..1519..30870..156065..252105..117649 ... Example of a hypertree with two hyperedges, one a 2-edge {3,4) and one a 3-edge{1,2,3}. ........__________........................ ......./..........\.______................ ......|....1...../.\......\............... ......|.........|.3.|....4.|.............. ......|....2.....\./______/............... .......\__________/....................... .......................................... 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: {1,2,3} and {3,4); {1,2,3} and {2,4); {1,2,3} and {1,4); {1,2,4} and {1,3); {1,2,4} and {2,3); {1,2,4} and {3,4); {1,3,4} and {1,2); {1,3,4} and {2,3); {1,3,4} and {2,4); {2,3,4} and {1,2); {2,3,4} and {1,3); {2,3,4} and {1,4). Choosing one of the four vertices as the root gives a total of 4x12 = 48 rooted hypertrees on 4 vertices. MAPLE with(combinat): A210586 := (n, k) -> n^k*stirling2(n-1, k): for n from 2 to 10 do seq(A210586(n, k), k = 1..n-1) end do; # Peter Bala, Oct 28 2015 CROSSREFS A035051 (row sums). Cf. A210587, A048993. Sequence in context: A249824 A227912 A229212 * A202017 A127198 A065631 Adjacent sequences:  A210583 A210584 A210585 * A210587 A210588 A210589 KEYWORD nonn,easy,tabl AUTHOR Peter Bala, Mar 26 2012 STATUS approved

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