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A058877
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Number of labeled acyclic digraphs with n nodes containing exactly n-1 points of in-degree zero.
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22
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0, 2, 9, 28, 75, 186, 441, 1016, 2295, 5110, 11253, 24564, 53235, 114674, 245745, 524272, 1114095, 2359278, 4980717, 10485740, 22020075, 46137322, 96468969, 201326568, 419430375, 872415206, 1811939301, 3758096356, 7784628195, 16106127330, 33285996513
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OFFSET
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1,2
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COMMENTS
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Let Q be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all nonempty elements x,y of P(A), xRy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y. Then a(n) = |Q|. - Ross La Haye, Feb 20 2008, Oct 21 2008
The La Haye binary relation Q is more clearly stated as x is nonempty and y has one more element than x. If x is a k-set than the number of such pairs is binomial( n, k) * (n-k). - Michael Somos, Mar 29 2012
Select one of the n nodes of the digraph and select a nonempty subset of the rest to connect to the selected node. This can be done in n * (2^(n-1) - 1) ways. - Michael Somos, Mar 29 2012
a(n) is the number of ternary sequences of length n that contain one 0 and at least one 1. For example, a(3)=9 since the sequences are the 3 permutations of 011 and the 6 permutations of 012. - Enrique Navarrete, Apr 05 2021
a(n) is also the number of multiplications required to compute the permanent of general n X n matrices using canonical trellis method (see Theorem 5, p. 10 in Kiah et al.). - Stefano Spezia, Nov 02 2021
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REFERENCES
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F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 19, (1.6.4).
Gerta Rucker and Christoph Rucker, "Walk counts, Labyrinthicity and complexity of acyclic and cyclic graphs and molecules", J. Chem. Inf. Comput. Sci., 40 (2000), 99-106. See Table 1 on page 101. [From Parthasarathy Nambi, Sep 26 2008]
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LINKS
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FORMULA
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G.f.: x^2*(2-3*x)/((1-2*x)*(1-x))^2.
a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4). (End)
a(n) = Sum_{k=1..n-1} binomial(n, k) * (n-k). - Michael Somos, Mar 29 2012
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EXAMPLE
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G.f. = 2*x^2 + 9*x^3 + 28*x^4 + 75*x^5 + 186*x^6 + 441*x^7 + 1016*x^8 + ...
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MAPLE
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a:=n->sum(k*binomial(n, k), k=2..n): seq(a(n), n=1..29); # Zerinvary Lajos, May 08 2007
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MATHEMATICA
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a[ n_] := Sum[ Binomial[ n, k] (n - k), {k, n-1}]; (* Michael Somos, Mar 29 2012 *)
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PROG
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(Sage) [stirling_number2(i, 2)*i for i in range(1, 26)] # Zerinvary Lajos, Jun 27 2008
(Sage) [(n+1)*gaussian_binomial(n, 1, 2) for n in range(0, 29)] # Zerinvary Lajos, May 31 2009
(PARI) {a(n) = if( n<1, 0, n * (2^(n-1) - 1))} /* Michael Somos, Mar 29 2012 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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