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A297568 Number of nonisomorphic proper colorings of partition star graph using four colors. 7
4, 12, 24, 36, 40, 108, 108, 60, 216, 180, 324, 324, 84, 360, 540, 648, 972, 972, 972, 112, 540, 1080, 660, 1080, 2916, 1512, 1944, 2916, 2916, 2916, 144, 756, 1800, 1980, 1620, 5832, 4860, 4536, 3240, 8748, 8748, 5832, 8748, 8748, 8748, 180, 1008, 2700, 3960, 1980, 2268, 9720, 14580, 9072, 13608, 4860, 17496, 14580, 26244, 13284, 9720, 26244, 26244, 17496, 26244, 26244, 26244, 220, 1296, 3780, 6600, 5940, 3024, 14580 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
A partition star graph consists of a multiset of paths with lengths given by the elements of the partition attached to a distinguished root node. The ordering of the partitions is by traversing antichains in Young's lattice bottom to top, left to right. Isomorphism refers to the automorphisms of the star graph corresponding to the partition.
LINKS
Marko Riedel et al., Orbital chromatic polynomials
FORMULA
For a partition lambda we have the OCP: k Product_{p^v in lambda} C((k-1)^p+v-1, v). Here we have k=4.
EXAMPLE
Rows are:
4;
12;
24, 36;
40, 108, 108;
60, 216, 180, 324, 324;
84, 360, 540, 648, 972, 972, 972;
MAPLE
b:= (n, i)-> `if`(n=0, [4], `if`(i<1, [], [seq(map(x-> x*
binomial(3^i+j-1, j), b(n-i*j, i-1))[], j=0..n/i)])):
T:= n-> b(n$2)[]:
seq(T(n), n=0..10); # Alois P. Heinz, Jan 14 2018
MATHEMATICA
b[n_, i_] := If[n == 0, {4}, If[i<1, {}, Table[Map[Function[x, x*Binomial[ 3^i + j - 1, j]], b[n - i*j, i - 1]], {j, 0, n/i}]] // Flatten];
T[n_] := b[n, n];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 17 2018, after Alois P. Heinz *)
CROSSREFS
Row sums give 4*A144067.
Row lengths give A000041.
Sequence in context: A285350 A072389 A218391 * A353795 A317518 A307763
KEYWORD
nonn,tabf
AUTHOR
Marko Riedel, Dec 31 2017
STATUS
approved

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Last modified May 19 00:35 EDT 2024. Contains 372666 sequences. (Running on oeis4.)