cp's OEIS Frontend

A297568 Number of nonisomorphic proper colorings of partition star graph using four colors.

%I A297568 #23 Jan 17 2018 13:15:34
%S A297568 4,12,24,36,40,108,108,60,216,180,324,324,84,360,540,648,972,972,972,
%T A297568 112,540,1080,660,1080,2916,1512,1944,2916,2916,2916,144,756,1800,
%U A297568 1980,1620,5832,4860,4536,3240,8748,8748,5832,8748,8748,8748,180,1008,2700,3960,1980,2268,9720,14580,9072,13608,4860,17496,14580,26244,13284,9720,26244,26244,17496,26244,26244,26244,220,1296,3780,6600,5940,3024,14580
%N A297568 Number of nonisomorphic proper colorings of partition star graph using four colors.
%C A297568 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 A297568 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2584928/">Orbital chromatic polynomials</a>
%F A297568 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=4.
%e A297568 Rows are:
%e A297568    4;
%e A297568   12;
%e A297568   24,  36;
%e A297568   40, 108, 108;
%e A297568   60, 216, 180, 324, 324;
%e A297568   84, 360, 540, 648, 972, 972, 972;
%p A297568 b:= (n, i)-> `if`(n=0, [4], `if`(i<1, [], [seq(map(x-> x*
%p A297568      binomial(3^i+j-1, j), b(n-i*j, i-1))[], j=0..n/i)])):
%p A297568 T:= n-> b(n$2)[]:
%p A297568 seq(T(n), n=0..10);  # _Alois P. Heinz_, Jan 14 2018
%t A297568 b[n_, i_] := If[n == 0, {4}, If[i<1, {}, Table[Map[Function[x, x*Binomial[ 3^i + j - 1, j]], b[n - i*j, i - 1]], {j, 0, n/i}]] // Flatten];
%t A297568 T[n_] := b[n, n];
%t A297568 Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jan 17 2018, after _Alois P. Heinz_ *)
%Y A297568 Cf. A297567, A297569, A297570.
%Y A297568 Row sums give 4*A144067.
%Y A297568 Row lengths give A000041.
%K A297568 nonn,tabf
%O A297568 0,1
%A A297568 _Marko Riedel_, Dec 31 2017