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A188674 Stack polyominoes with square core. 11
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: (Start)

Also the number of integer partitions of n with final part in their inner lining partition equal to 1. The k-th part of the inner lining partition of an integer 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(11) = 13 partitions are:

  (22)  (32)   (42)    (52)     (62)      (72)       (82)        (92)

        (221)  (321)   (421)    (521)     (333)      (433)       (443)

               (2211)  (3211)   (4211)    (621)      (721)       (533)

                       (22111)  (32111)   (5211)     (3331)      (821)

                                (221111)  (42111)    (6211)      (3332)

                                          (321111)   (52111)     (4331)

                                          (2211111)  (421111)    (7211)

                                                     (3211111)   (33311)

                                                     (22111111)  (62111)

                                                                 (521111)

                                                                 (4211111)

                                                                 (32111111)

                                                                 (221111111)

(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, A115994, A252464, A257990, A325135, A325163, A325165.

Sequence in context: A333265 A055167 A064628 * A320316 A236166 A017834

Adjacent sequences:  A188671 A188672 A188673 * A188675 A188676 A188677

KEYWORD

nonn

AUTHOR

Emanuele Munarini, Apr 08 2011

STATUS

approved

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Last modified February 27 20:40 EST 2021. Contains 341658 sequences. (Running on oeis4.)