A298263 Number of nonisomorphic proper colorings of partition multicycle graph using three colors.

1, 3, 6, 3, 10, 9, 2, 15, 18, 6, 6, 6, 21, 30, 18, 12, 6, 18, 6, 28, 45, 36, 10, 20, 18, 3, 36, 18, 18, 14, 36, 63, 60, 30, 30, 36, 12, 9, 60, 54, 12, 36, 18, 42, 18, 45, 84, 90, 60, 15, 42, 60, 36, 18, 9, 90, 108, 36, 36, 21, 60, 54, 12, 84, 42, 54, 36, 55, 108, 126, 100, 45, 56, 90, 72, 20, 30, 27, 4, 126, 180, 108, 72, 36, 63, 90, 108, 36, 36, 36, 140, 126, 28, 108, 54, 108, 58

Offset: 0

Author

Marko Riedel, Jan 15 2018

Keywords

• nonn
• tabf

Comments

A partition multicycle graph consists of a multiset of cycles with lengths given by the elements of the partition where degenerate cycles on one node are taken to be singletons and on two nodes a pair of nodes connected by an edge. 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 multicycle graph corresponding to the partition, consisting of permutations of cycles of the same length combined with rotations of individual cycles (no dihedral symmetry).

Examples

			Rows are:
   1;
   3;
   6;
  10,  9,  2;
  15, 18,  6,  6, 6;
  21, 30, 18, 12, 6, 18, 6;
		

Links

• Marko Riedel et al., Orbital chromatic polynomials
• Marko Riedel, Maple code computing OCP for sequences A298263, A298264, A298265, A298266.

Crossrefs

Cf. A297567, A297568, A297569, A297570, A298264, A298265, A298266.

Formula

For a partition lambda we have the OCP: Product_{p^v in lambda} C(Q_p(k)+v-1, v)

where Q_1(k) = k, Q_2(k) = k(k-1)/2 and for n>=3, Q_n(k) = (1/n) * Sum_{d|n} phi(n/d) P_d(k) with P_d(k) = (k-1)^d + (-1)^d (k-1). Here we have k=3.