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A320653 a(n) is the associated coefficient of the n-uniform simplex. 0
2, 21, 588, 28230, 2092206, 220611384, 31373370936, 5785037767440, 1342136211324090, 382559909729171328, 131411551493995125828, 53537846795391076075776, 25523603120175022166538150, 14076445847378724286239575040, 8892219411843450738850246324464 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

The associated coefficient of a hypergraph is the weight assigned to it in the generalized Harary-Sachs formula (arxiv link to be posted).  For example, the 2-uniform simplex is a triangle and a(2) = 2.  Famously the codegree 3 coefficient of the adjacency characteristic polynomial of a graph is -a(2)(# of triangles in G).  The quantity a(n)=C_n as mentioned in Cooper and Dutle wherein the authors computed the values up to n=5.

LINKS

Table of n, a(n) for n=2..16.

Gregory Clark, Joshua Cooper, A Harary-Sachs Theorem for Hypergraphs, arXiv:1812.00468 [math.CO], 2018.

J. Cooper, A. Dutle, Spectra of uniform hypergraphs, Linear Algebra Appl. 436 (2012) 3268-3292.

FORMULA

Let P(n,2) denote the set of partitions of n where each part is of size at least 2.  Let L(p) denote the length of p, let p(i) denote the size of part i of p, and let V(p,i) denote the number of parts of p which have size i.  For p in P(n,2) let f(n,p)= n!/((Product_{i=1..L(p)} p(i))(Product_{i=2..n}V(p,i)).  Then a(n) = (1/((n-1)*(n+1)^2))*Sum_{p in P(n+1,2)}(f(n+1,p)*Product_{i=1 .. L(p)}(n^(p(i)) + (-1)^(p(i)+1))).

a(n) = exp(n*log(n)(2+o(1)).

PROG

(Sage)

def simplex_coefficient(n):

    P=Partitions(n+1, min_part=2)

    x=0

    for p in P:

        E = p.evaluation()

        tau=1

        c = 1

        d=1

        for i in p:

            tau = tau*(n^i+(-1)^(i+1))

            c = c * i

        for i in E:

            d = d * factorial(i)

        x= x + tau/(c * d)

    return factorial(n+1)*x/((n-1)*(n+1)^2)

[simplex_coefficient(n) for n in range(2, 4)]

CROSSREFS

Sequence in context: A171107 A218768 A195736 * A302686 A078602 A060319

Adjacent sequences:  A320650 A320651 A320652 * A320654 A320655 A320656

KEYWORD

nonn

AUTHOR

Gregory J. Clark, Oct 18 2018

STATUS

approved

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Last modified July 28 15:51 EDT 2021. Contains 346335 sequences. (Running on oeis4.)