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%I #23 May 23 2022 17:46:48
%S 1,1,0,0,1,2,3,4,5,7,9,13,17,24,31,42,54,71,90,117,147,188,236,298,
%T 371,466,576,716,882,1088,1331,1633,1987,2422,2935,3557,4290,5177,
%U 6216,7465,8932,10682,12731,15169,18016,21387,25321,29955,35353,41696,49063,57689,67698,79375,92896,108633,126817,147922,172272
%N Stack polyominoes with square core.
%C a(n) is the number of stack polyominoes of area n with square core.
%C The core of stack is the set of all maximal columns.
%C The core is a square when the number of columns is equal to their height.
%C 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:
%C (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).
%C From _Gus Wiseman_, Apr 06 2019 and May 21 2022: (Start)
%C 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:
%C (22) (32) (42) (52) (62) (72) (82)
%C (221) (321) (421) (521) (333) (433)
%C (2211) (3211) (4211) (621) (721)
%C (22111) (32111) (5211) (3331)
%C (221111) (42111) (6211)
%C (321111) (52111)
%C (2211111) (421111)
%C (3211111)
%C (22111111)
%C 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:
%C () . . (21) (31) (41) (51) (61) (71)
%C (211) (311) (411) (511) (332)
%C (2111) (3111) (4111) (611)
%C (21111) (31111) (5111)
%C (211111) (41111)
%C (311111)
%C (2111111)
%C Also partitions of n + 1 without a fixed point or conjugate fixed point.
%C (End)
%H Alois P. Heinz, <a href="/A188674/b188674.txt">Table of n, a(n) for n = 0..1000</a>
%H Brian Hopkins, James A. Sellers, and Dennis Stanton, <a href="https://arxiv.org/abs/2009.10873">Dyson's Crank and the Mex of Integer Partitions</a>, arXiv:2009.10873 [math.CO], 2020. Mentions this sequence.
%F G.f.: 1 + sum(k>=0, x^((k+1)^2)/((1-x)^2*(1-x^2)^2*...*(1-x^k)^2)).
%t 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]
%t (* second program *)
%t pml[ptn_]:=If[ptn=={},{},FixedPointList[If[#=={},{},DeleteCases[Rest[#]-1,0]]&,ptn][[-3]]];
%t Table[Length[Select[IntegerPartitions[n],pml[#]=={1}&]],{n,0,30}] (* _Gus Wiseman_, Apr 06 2019 *)
%Y Cf. A001523 (stacks).
%Y Cf. A006918, A252464, A325135, A325163, A325165.
%Y Positive crank: A001522, ranked by A352874.
%Y Zero crank: A064410, ranked by A342192.
%Y Nonnegative crank: A064428, ranked by A352873.
%Y Fixed point but no conjugate fixed point: A118199, ranked by A353316.
%Y A000041 counts partitions, strict A000009.
%Y A002467 counts permutations with a fixed point, complement A000166.
%Y A115720/A115994 count partitions by Durfee square, rank statistic A257990.
%Y A238352 counts reversed partitions by fixed points, rank statistic A352822.
%Y A238394 counts reversed partitions without a fixed point, ranked by A352830.
%Y A238395 counts reversed partitions with a fixed point, ranked by A352872.
%Y A352833 counts partitions by fixed points.
%Y Cf. A000700, A088902, A114088, A300788, A330644, A352828, A352829, A352832.
%K nonn
%O 0,6
%A _Emanuele Munarini_, Apr 08 2011
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