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

%I A298263 #12 Jan 22 2018 18:46:31
%S A298263 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,
%T A298263 18,18,14,36,63,60,30,30,36,12,9,60,54,12,36,18,42,18,45,84,90,60,15,
%U A298263 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
%N A298263 Number of nonisomorphic proper colorings of partition multicycle graph using three colors.
%C A298263 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 A298263 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2584928/">Orbital chromatic polynomials</a>
%H A298263 Marko Riedel, <a href="/A298263/a298263.maple.txt">Maple code computing OCP for sequences A298263, A298264, A298265, A298266.</a>
%F A298263 For a partition lambda we have the OCP: Product_{p^v in lambda} C(Q_p(k)+v-1, v)
%F A298263 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.
%e A298263 Rows are:
%e A298263    1;
%e A298263    3;
%e A298263    6;
%e A298263   10,  9,  2;
%e A298263   15, 18,  6,  6, 6;
%e A298263   21, 30, 18, 12, 6, 18, 6;
%Y A298263 Cf. A297567, A297568, A297569, A297570, A298264, A298265, A298266.
%K A298263 nonn,tabf
%O A298263 0,2
%A A298263 _Marko Riedel_, Jan 15 2018