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A188674
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Stack polyominoes with square core.
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21
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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
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OFFSET
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0,6
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COMMENTS
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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).
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)
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LINKS
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FORMULA
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G.f.: 1 + sum(k>=0, x^((k+1)^2)/((1-x)^2*(1-x^2)^2*...*(1-x^k)^2)).
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MATHEMATICA
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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 *)
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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