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
Gregory Clark and Joshua Cooper, A Harary-Sachs Theorem for Hypergraphs, arXiv:1812.00468 [math.CO], 2018.
Gregory J. Clark and Joshua Cooper, Applications of the Harary-Sachs Theorem for Hypergraphs, arXiv:2107.10781 [math.CO], 2021.
J. Cooper and 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
KEYWORD
nonn
AUTHOR
Gregory J. Clark, Oct 18 2018
STATUS
approved