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

3, 6, 9, 12, 12, 24, 24, 15, 36, 30, 48, 48, 18, 48, 60, 72, 96, 96, 96, 21, 60, 90, 60, 96, 192, 108, 144, 192, 192, 192, 24, 72, 120, 120, 120, 288, 240, 216, 192, 384, 384, 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

Offset

0,1

Comments

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.

Links

Table of n, a(n) for n=0..90.

Marko Riedel et al., Orbital chromatic polynomials

Marko Riedel, Maple code computing OCP for sequences A297567, A297568, A297569, A297570.

Formula

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.

Example

Rows are:
   3;
   6;
   9, 12;
  12, 24, 24;
  15, 36, 30, 48, 48;
  18, 48, 60, 72, 96, 96, 96;

Maple

b:= (n, i)-> `if`(n=0, [3], `if`(i<1, [], [seq(map(x-> x*
     binomial(2^i+j-1, j), b(n-i*j, i-1))[], j=0..n/i)])):
T:= n-> b(n$2)[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Jan 14 2018

Mathematica

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[n_] :=  b[n, n];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 17 2018, after Alois P. Heinz *)

Crossrefs

Cf. A297568, A297569, A297570.

Row sums give 3*A034899.

Row lengths give A000041.

Sequence in context: A122809 A066662 A329517 * A285402 A153403 A336341

Adjacent sequences: A297564 A297565 A297566 * A297568 A297569 A297570

Keyword

nonn,tabf

Author

Marko Riedel, Dec 31 2017

Status

approved