A297567 - OEIS

%I #24 Jan 17 2018 13:14:45

%S 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 192,108,144,192,192,192,24,72,120,120,120,288,240,216,192,384,384,

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

%C 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 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2584928/">Orbital chromatic polynomials</a>

%H Marko Riedel, <a href="/A297567/a297567.maple.txt">Maple code computing OCP for sequences A297567, A297568, A297569, A297570. </a>

%F 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 Rows are:

%e    3;

%e    6;

%e    9, 12;

%e   12, 24, 24;

%e   15, 36, 30, 48, 48;

%e   18, 48, 60, 72, 96, 96, 96;

%p b:= (n, i)-> `if`(n=0, [3], `if`(i<1, [], [seq(map(x-> x*

%p      binomial(2^i+j-1, j), b(n-i*j, i-1))[], j=0..n/i)])):

%p T:= n-> b(n$2)[]:

%p seq(T(n), n=0..10);  # _Alois P. Heinz_, Jan 14 2018

%t 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 T[n_] :=  b[n, n];

%t Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jan 17 2018, after _Alois P. Heinz_ *)

%Y Cf. A297568, A297569, A297570.

%Y Row sums give 3*A034899.

%Y Row lengths give A000041.

%K nonn,tabf

%O 0,1

%A _Marko Riedel_, Dec 31 2017