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

Offset: 0

Author

Emanuele Munarini, Apr 08 2011

Keywords

• nonn

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.

Crossrefs

Cf. A001523 (stacks).

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

Positive crank: A001522, ranked by A352874.

Zero crank: A064410, ranked by A342192.

Nonnegative crank: A064428, ranked by A352873.

Fixed point but no conjugate fixed point: A118199, ranked by A353316.

A000041 counts partitions, strict A000009.

A002467 counts permutations with a fixed point, complement A000166.

A115720/A115994 count partitions by Durfee square, rank statistic A257990.

A238352 counts reversed partitions by fixed points, rank statistic A352822.

A238394 counts reversed partitions without a fixed point, ranked by A352830.

A238395 counts reversed partitions with a fixed point, ranked by A352872.

A352833 counts partitions by fixed points.

Cf. A000700, A088902, A114088, A300788, A330644, A352828, A352829, A352832.

Programs

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 *)

Formula

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