A115868 Invariants for a hidden action of S_(n+1) on Cayley trees with n vertices.

1, 1, 1, 1, 2, 2, 4, 6, 11, 18, 39, 70, 153, 321, 721, 1612, 3792, 8896, 21498, 52230, 128994, 320786, 806582, 2040912, 5205311, 13352470, 34460430, 89384609

Offset: 2

Author

F. Chapoton, Mar 14 2006

Keywords

• nonn
• more

Comments

This is the multiplicity of the trivial module in a sequence of modules of dimension (n-1)^(n-3) over the symmetric groups S_n. The restriction of these modules to S_(n-1) is given by the action on trees.

Examples

			M[6]=s[2, 1, 1, 1, 1] + 3 s[2, 2, 2] + 2 s[3, 1, 1, 1] + 2 s[3, 2, 1] + s[4, 1, 1] + 4 s[4, 2] + s[5, 1] + 2 s[6] as a sum of Schur functions hence a[6]=2.
		

Crossrefs

Cf. A000055 and A000272.

Cf. A000081, A051573, A098091.

Formula

No simple formula known, only a complicated sum over partitions.

It seems that a(n+1) = A000055(n) + A051573(n) - A000081(n). - Andrey Zabolotskiy, Aug 05 2024

Extensions

Five more terms added by F. Chapoton, Mar 08 2020