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

4, 12, 24, 36, 40, 108, 108, 60, 216, 180, 324, 324, 84, 360, 540, 648, 972, 972, 972, 112, 540, 1080, 660, 1080, 2916, 1512, 1944, 2916, 2916, 2916, 144, 756, 1800, 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

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..73.

Marko Riedel et al., Orbital chromatic polynomials

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=4.

Example

Rows are:
   4;
  12;
  24,  36;
  40, 108, 108;
  60, 216, 180, 324, 324;
  84, 360, 540, 648, 972, 972, 972;

Maple

b:= (n, i)-> `if`(n=0, [4], `if`(i<1, [], [seq(map(x-> x*
     binomial(3^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, {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[n_] := b[n, n];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 17 2018, after Alois P. Heinz *)

Crossrefs

Cf. A297567, A297569, A297570.

Row sums give 4*A144067.

Row lengths give A000041.

Sequence in context: A285350 A072389 A218391 * A353795 A317518 A307763

Adjacent sequences: A297565 A297566 A297567 * A297569 A297570 A297571

Keyword

nonn,tabf

Author

Marko Riedel, Dec 31 2017

Status

approved