A297568 Number of nonisomorphic proper colorings of partition star graph using four colors.

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

Offset: 0

Author

Marko Riedel, Dec 31 2017

Keywords

• nonn
• tabf

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.

Examples

			Rows are:
   4;
  12;
  24,  36;
  40, 108, 108;
  60, 216, 180, 324, 324;
  84, 360, 540, 648, 972, 972, 972;
		

Links

• Marko Riedel et al., Orbital chromatic polynomials

Crossrefs

Cf. A297567, A297569, A297570.

Row sums give 4*A144067.

Row lengths give A000041.

Programs

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 *)

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.