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A118199
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Number of partitions of n having no parts equal to the size of their Durfee squares.
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12
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1, 0, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 31, 40, 53, 68, 89, 113, 146, 184, 234, 293, 369, 458, 572, 706, 874, 1073, 1320, 1611, 1970, 2393, 2909, 3518, 4255, 5122, 6167, 7394, 8862, 10585, 12637, 15038, 17886, 21213, 25141, 29723, 35112, 41383, 48737, 57278
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OFFSET
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0,7
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COMMENTS
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Also the number of integer partitions of n > 0 that have a fixed point but whose conjugate does not, ranked by A353316. For example, the a(5) = 1 through a(10) = 10 partitions are:
11111 222 322 422 522 622
111111 2221 2222 3222 4222
1111111 3221 4221 5221
22211 22221 22222
11111111 32211 32221
222111 42211
111111111 222211
322111
2221111
1111111111
Partitions w/o a fixed point or conjugate fixed point: A188674 (shifted).
(End)
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LINKS
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FORMULA
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G.f.: 1+sum(x^(k^2+k)/[(1-x^k)*product((1-x^i)^2, i=1..k-1)], k=1..infinity).
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EXAMPLE
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a(7) = 3 because we have [7] with size of Durfee square 1, [4,3] with size of Durfee square 2 and [3,3,1] with size of Durfee square 2.
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MAPLE
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g:=1+sum(x^(k^2+k)/(1-x^k)/product((1-x^i)^2, i=1..k-1), k=1..20): gser:=series(g, x=0, 60): seq(coeff(gser, x, n), n=0..54);
# second Maple program::
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
end:
a:= n-> add(add(b(k, d) *b(n-d*(d+1)-k, d-1),
k=0..n-d*(d+1)), d=0..floor(sqrt(n))):
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := Sum[Sum[b[k, d]*b[n-d*(d+1)-k, d-1], {k, 0, n-d*(d+1)}], {d, 0, Floor[Sqrt[n]]}]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)
pq[y_]:=Length[Select[Range[Length[y]], #==y[[#]]&]];
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Length[Select[IntegerPartitions[n], pq[#]>0&&pq[conj[#]]==0&]], {n, 0, 30}] (* a(0) = 0, Gus Wiseman, May 21 2022 *)
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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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