OFFSET
1,4
LINKS
Christian Barrientos, Sarah Minion, On the number of alpha-labeled graphs, Discussiones Mathematicae Graph Theory, to appear.
J. A. Gallian, A dynamic survey of graph labeling, Elec. J. Combin., (2013), #DS6.
David A. Sheppard, The factorial representation of major balanced labelled graphs, Discrete Math., 15(1976), no. 4, 379-388.
FORMULA
a(n) = Sum_{L=1..n-2} Sum_{i=1..n-1} Product_{k=1..n} d(L,k,i), where
for i < L,
d(L,k) if 1 <= k <= i,
d(L,k,i) ={ d(L,k) - 1 if i < k < n - i,
d(L,k) if n - i <= k <= n;
for i > L + 1,
d(L,k) if 1 <= k <= n - i,
d(L,k,i) ={ d(L,k) - 1 if n - i < k < n - i + L + 2,
d(L,k) if n - i + L + 2 <= k <= n.
k if 1 <= k < m,
d(L,k) ={ L + 1 if m <= k <= M,
n + 1 - k if M < k <= n,
m = min{L + 1, n - L}, M = max{L + 1, n - L}.
EXAMPLE
For n=4, a(4)=2, there are 2 alpha-labeled graphs with 4 edges and at most 4 vertices.
For n=10, a(10)=71944, there are 71944 alpha-labeled graphs with 10 edges and at most 10 vertices.
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Sarah Minion and Christian Barrientos, Jul 24 2014
STATUS
approved