cp's OEIS Frontend

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

%I A298264 #10 Sep 22 2020 05:32:59
%S A298264 1,4,10,6,20,24,8,35,60,21,32,24,56,120,84,80,48,96,48,84,210,210,56,
%T A298264 160,192,36,240,144,192,130,120,336,420,224,280,480,168,144,480,576,
%U A298264 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
%N A298264 Number of nonisomorphic proper colorings of partition multicycle graph using four colors.
%C A298264 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).
%H A298264 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2584928/">Orbital chromatic polynomials</a>
%F A298264 For a partition lambda we have the OCP: Product_{p^v in lambda} C(Q_p(k)+v-1, v)
%F A298264 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.
%e A298264 Rows are:
%e A298264    1;
%e A298264    4;
%e A298264   10,   6;
%e A298264   20,  24,  8;
%e A298264   35,  60, 21, 32, 24;
%e A298264   56, 120, 84, 80, 48, 96, 48;
%Y A298264 Cf. A297567, A297568, A297569, A297570, A298263, A298265, A298266.
%K A298264 nonn,tabf
%O A298264 0,2
%A A298264 _Marko Riedel_, Jan 15 2018