A188674 - OEIS

A188674

Stack polyominoes with square core.

1, 1, 0, 0, 1, 2, 3, 4, 5, 7, 9, 13, 17, 24, 31, 42, 54, 71, 90, 117, 147, 188, 236, 298, 371, 466, 576, 716, 882, 1088, 1331, 1633, 1987, 2422, 2935, 3557, 4290, 5177, 6216, 7465, 8932, 10682, 12731, 15169, 18016, 21387, 25321, 29955, 35353, 41696, 49063, 57689, 67698, 79375, 92896, 108633, 126817, 147922, 172272

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

a(n) is the number of stack polyominoes of area n with square core.

The core of stack is the set of all maximal columns.

The core is a square when the number of columns is equal to their height.

Equivalently, a(n) is the number of unimodal compositions of n, where the number of the parts of maximum value equal the maximum value itself. For instance, for n = 10, we have the following stacks:

(1,3,3,3), (3,3,3,1), (1,1,1,1,1,1,2,2), (1,1,1,1,1,2,2,1), (1,1,1,1,2,2,1,1), (1,1,1,2,2,1,1,1), (1,1,2,2,1,1,1,1), (1,2,2,1,1,1,1,1), (2,2,1,1,1,1,1,1).

From Gus Wiseman, Apr 06 2019 and May 21 2022: (Start)

Also the number of integer partitions of n with final part in their inner lining partition equal to 1, where the k-th part of the inner lining partition of a partition is the number of squares in its Young diagram that are k diagonal steps from the lower-right boundary. For example, the a(4) = 1 through a(10) = 9 partitions are:

(22) (32) (42) (52) (62) (72) (82)

(221) (321) (421) (521) (333) (433)

(2211) (3211) (4211) (621) (721)

(22111) (32111) (5211) (3331)

(221111) (42111) (6211)

(321111) (52111)

(2211111) (421111)

(3211111)

(22111111)

Also partitions that have a fixed point and a conjugate fixed point, ranked by A353317. The strict case is A352829. For example, the a(0) = 0 through a(9) = 7 partitions are:

() . . (21) (31) (41) (51) (61) (71)

(211) (311) (411) (511) (332)

(2111) (3111) (4111) (611)

(21111) (31111) (5111)

(211111) (41111)

(311111)

(2111111)

Also partitions of n + 1 without a fixed point or conjugate fixed point.

(End)

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020. Mentions this sequence.

FORMULA

G.f.: 1 + sum(k>=0, x^((k+1)^2)/((1-x)^2*(1-x^2)^2*...*(1-x^k)^2)).

MATHEMATICA

a[n_]:=CoefficientList[Series[1+Sum[x^((k+1)^2)/Product[(1-x^i)^2, {i, 1, k}], {k, 0, n}], {x, 0, n}], x]

(* second program *)

pml[ptn_]:=If[ptn=={}, {}, FixedPointList[If[#=={}, {}, DeleteCases[Rest[#]-1, 0]]&, ptn][[-3]]];

Table[Length[Select[IntegerPartitions[n], pml[#]=={1}&]], {n, 0, 30}] (* Gus Wiseman, Apr 06 2019 *)

CROSSREFS

Cf. A001523 (stacks).

Cf. A006918, A252464, A325135, A325163, A325165.

Positive crank: A001522, ranked by A352874.

Zero crank: A064410, ranked by A342192.