A298264 Number of nonisomorphic proper colorings of partition multicycle graph using four colors.

1, 4, 10, 6, 20, 24, 8, 35, 60, 21, 32, 24, 56, 120, 84, 80, 48, 96, 48, 84, 210, 210, 56, 160, 192, 36, 240, 144, 192, 130, 120, 336, 420, 224, 280, 480, 168, 144, 480, 576, 192, 480, 288, 520, 312, 165, 504, 735, 560, 126, 448, 960, 672, 360, 216, 840, 1440, 504, 768, 300, 960, 1152, 384, 1300, 780, 1248, 834, 220, 720, 1176, 1120, 504, 672, 1680, 1680, 448, 720, 864, 120, 1344, 2880, 2016, 1920, 1152, 1200, 1680, 2880, 1008, 1536, 1152, 2600, 3120, 1040, 3120, 1872, 3336, 2192

Offset

0,2

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).

Links

Table of n, a(n) for n=0..96.

Marko Riedel et al., Orbital chromatic polynomials

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=4.

Example

Rows are:
   1;
   4;
  10,   6;
  20,  24,  8;
  35,  60, 21, 32, 24;
  56, 120, 84, 80, 48, 96, 48;

Crossrefs

Cf. A297567, A297568, A297569, A297570, A298263, A298265, A298266.

Sequence in context: A205016 A241619 A129531 * A014476 A080362 A365781

Adjacent sequences: A298261 A298262 A298263 * A298265 A298266 A298267

Keyword

nonn,tabf

Author

Marko Riedel, Jan 15 2018

Status

approved