cp's OEIS Frontend

A297567 Number of nonisomorphic proper colorings of partition star graph using three colors.

%I A297567 #24 Jan 17 2018 13:14:45
%S A297567 3,6,9,12,12,24,24,15,36,30,48,48,18,48,60,72,96,96,96,21,60,90,60,96,
%T A297567 192,108,144,192,192,192,24,72,120,120,120,288,240,216,192,384,384,
%U A297567 288,384,384,384,27,84,150,180,105,144,384,480,324,432,240,576,480,768,408,384,768,768,576,768,768,768,30,96,180,240,210,168,480,720,480,432,864,360,288,768,960,1152,1536,816,480,1152,960,1536,1536,768
%N A297567 Number of nonisomorphic proper colorings of partition star graph using three colors.
%C A297567 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.
%H A297567 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2584928/">Orbital chromatic polynomials</a>
%H A297567 Marko Riedel, <a href="/A297567/a297567.maple.txt">Maple code computing OCP for sequences A297567, A297568, A297569, A297570. </a>
%F A297567 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=3.
%e A297567 Rows are:
%e A297567    3;
%e A297567    6;
%e A297567    9, 12;
%e A297567   12, 24, 24;
%e A297567   15, 36, 30, 48, 48;
%e A297567   18, 48, 60, 72, 96, 96, 96;
%p A297567 b:= (n, i)-> `if`(n=0, [3], `if`(i<1, [], [seq(map(x-> x*
%p A297567      binomial(2^i+j-1, j), b(n-i*j, i-1))[], j=0..n/i)])):
%p A297567 T:= n-> b(n$2)[]:
%p A297567 seq(T(n), n=0..10);  # _Alois P. Heinz_, Jan 14 2018
%t A297567 b[n_, i_] := If[n == 0, {3}, If[i<1, {}, Table[Map[Function[x, x*Binomial[ 2^i + j - 1, j]], b[n - i*j, i - 1]], {j, 0, n/i}]] // Flatten];
%t A297567 T[n_] :=  b[n, n];
%t A297567 Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jan 17 2018, after _Alois P. Heinz_ *)
%Y A297567 Cf. A297568, A297569, A297570.
%Y A297567 Row sums give 3*A034899.
%Y A297567 Row lengths give A000041.
%K A297567 nonn,tabf
%O A297567 0,1
%A A297567 _Marko Riedel_, Dec 31 2017