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A355226
Irregular triangle read by rows where T(n,k) is the number of independent sets of size k in the n-halved cube graph.
2
1, 1, 1, 2, 1, 4, 1, 8, 4, 1, 16, 40, 1, 32, 256, 480, 120, 1, 64, 1344, 11200, 36400, 40320, 13440, 1920, 240, 1, 128, 6336, 156800, 2104480, 15644160, 63672000, 136970880, 147748560, 76396800, 21087360, 4273920, 840000, 161280, 28800, 3840, 240
OFFSET
1,4
COMMENTS
The independence number alpha(G) of a graph is the cardinality of the largest independent vertex set. The n-halved graph has alpha(G) = A005864(n). The independence polynomial for the n-halved cube is given by Sum_{k=0..alpha(G)} T(n,k)*t^k.
Since 0 <= k <= alpha(G), row n has length A005864(n) + 1.
LINKS
Eric Weisstein's World of Mathematics, Independence polynomial
Eric Weisstein's World of Mathematics, Halved cube graph
EXAMPLE
Triangle begins:
k = 0 1 2
n = 1: 1, 1
n = 2: 1, 2
n = 3: 1, 4
n = 4: 1, 8, 4
n = 5: 1, 16, 40
The 4-halved cube graph has independence polynomial 1 + 8*t + 4*t^2.
PROG
(Sage) from sage.graphs.independent_sets import IndependentSets
from collections import Counter
def row(n):
if n == 1:
g = graphs.CompleteGraph(1)
else:
g = graphs.HalfCube(n)
setCounts = Counter()
for Iset in IndependentSets(g):
setCounts[len(Iset)] += 1
outList = [0] * len(setCounts)
for n in range(0, len(setCounts)):
outList[n] = setCounts[n]
return outList
CROSSREFS
Row sums are A288943.
Sequence in context: A146679 A146221 A147457 * A107061 A112481 A134851
KEYWORD
nonn,tabf
AUTHOR
Christopher Flippen, Jun 24 2022
STATUS
approved