cp's OEIS Frontend

A297570 Number of nonisomorphic proper colorings of partition star graph using six colors.

%I A297570 #26 Sep 22 2020 05:32:49
%S A297570 6,30,90,150,210,750,750,420,2250,1950,3750,3750,756,5250,9750,11250,
%T A297570 18750,18750,18750,1260,10500,29250,17550,26250,93750,47250,56250,
%U A297570 93750,93750,93750,1980,18900,68250,87750,52500,281250,243750,236250,131250,468750,468750,281250,468750,468750,468750,2970,31500,136500,263250,122850,94500,656250,1218750,708750,1181250,262500,1406250,1218750,2343750,1173750
%N A297570 Number of nonisomorphic proper colorings of partition star graph using six colors.
%C A297570 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 A297570 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2584928/">Orbital chromatic polynomials</a>
%F A297570 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=6.
%e A297570 Rows are:
%e A297570     6;
%e A297570    30;
%e A297570    90,  150;
%e A297570   210,  750,  750;
%e A297570   420, 2250, 1950,  3750,  3750;
%e A297570   756, 5250, 9750, 11250, 18750, 18750, 18750;
%p A297570 b:= (n, i)-> `if`(n=0, [6], `if`(i<1, [], [seq(map(x-> x*
%p A297570      binomial(5^i+j-1, j), b(n-i*j, i-1))[], j=0..n/i)])):
%p A297570 T:= n-> b(n$2)[]:
%p A297570 seq(T(n), n=0..10);  # _Alois P. Heinz_, Jan 14 2018
%t A297570 b[n_, i_] := If[n == 0, {6}, If[i<1, {}, Table[Map[Function[x, x*Binomial[ 5^i + j - 1, j]], b[n - i*j, i - 1]], {j, 0, n/i}]] // Flatten];
%t A297570 T[n_] :=  b[n, n];
%t A297570 Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jan 17 2018, after _Alois P. Heinz_ *)
%Y A297570 Cf. A297567, A297568, A297569.
%Y A297570 Row sums give 6*A144069.
%Y A297570 Row lengths give A000041.
%K A297570 nonn,tabf
%O A297570 0,1
%A A297570 _Marko Riedel_, Dec 31 2017