A096577 - OEIS

A096577

Number of fixed points of solid partitions under 'time-lapse' operation.

1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 4, 1, 5, 0, 7, 1

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OFFSET

1,9

COMMENTS

Operation 'time lapse', or 'lapse', L, operates on a solid partition by creating a new one, layer by layer. Layer k is defined by its 3-dimensional-Ferrers plot, equal to the (existence of) elements of the solid partition with value >= k. As if taking a time-lapse picture of the solid partition, filtering out elements less than k and projecting the resulting structure (filled with ones) to the base plane. Given there are three planes to project into, together with the starting solid partition, that makes four 'isomers'.

LINKS

Table of n, a(n) for n=1..20.

Wouter Meeussen, Mma functions for plane and solid partitions

EXAMPLE

Solid partition [{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}] lapses (L) into

[{{4,1},{2},{1},{1},{1}},{{1,1},{1}},{{1,1}}], then into

[{{2,1,1,1,1},{2,1},{2}},{{1,1}},{{1}},{{1}}], further into

[{{5,2,1},{2},{1},{1}},{{1,1,1}}] and returns after L^4 to

[{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}].

MATHEMATICA

(* See link above. *)

Tr/@Table[Count[solidformBTK[par], arg_z/; lapse[arg]==arg], {n, 20}, {par, IntegerPartitions[n]}]. (* Wouter Meeussen, Feb 05 2025 *)