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%I #21 May 23 2022 17:46:39
%S 1,0,1,1,1,1,2,3,5,7,10,13,18,23,31,40,53,68,89,113,146,184,234,293,
%T 369,458,572,706,874,1073,1320,1611,1970,2393,2909,3518,4255,5122,
%U 6167,7394,8862,10585,12637,15038,17886,21213,25141,29723,35112,41383,48737,57278
%N Number of partitions of n having no parts equal to the size of their Durfee squares.
%C a(n) = A118198(n,0).
%C From _Gus Wiseman_, May 21 2022: (Start)
%C 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:
%C 11111 222 322 422 522 622
%C 111111 2221 2222 3222 4222
%C 1111111 3221 4221 5221
%C 22211 22221 22222
%C 11111111 32211 32221
%C 222111 42211
%C 111111111 222211
%C 322111
%C 2221111
%C 1111111111
%C Partitions w/ a fixed point: A001522 (unproved), ranked by A352827 (cf. A352874).
%C Partitions w/o a fixed point: A064428 (unproved), ranked by A352826 (cf. A352873).
%C Partitions w/ a fixed point and a conjugate fixed point: A188674, reverse A325187, ranked by A353317.
%C Partitions w/o a fixed point or conjugate fixed point: A188674 (shifted).
%C (End)
%H Alois P. Heinz, <a href="/A118199/b118199.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: 1+sum(x^(k^2+k)/[(1-x^k)*product((1-x^i)^2, i=1..k-1)], k=1..infinity).
%e 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.
%p 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);
%p # second Maple program::
%p b:= proc(n, i) option remember; `if`(n=0, 1,
%p `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
%p end:
%p a:= n-> add(add(b(k, d) *b(n-d*(d+1)-k, d-1),
%p k=0..n-d*(d+1)), d=0..floor(sqrt(n))):
%p seq(a(n), n=0..70); # _Alois P. Heinz_, Apr 09 2012
%t 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_ *)
%t pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t Table[Length[Select[IntegerPartitions[n],pq[#]>0&&pq[conj[#]]==0&]],{n,0,30}] (* a(0) = 0, _Gus Wiseman_, May 21 2022 *)
%Y Column k=0 of A118198.
%Y A000041 counts partitions, strict A000009.
%Y A000700 = self-conjugate partitions, ranked by A088902, complement A330644.
%Y A002467 counts permutations with a fixed point, complement A000166.
%Y A064410 counts partitions of crank 0, ranked by A342192.
%Y A115720 and A115994 count partitions by Durfee square, rank stat 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. A114088, A300788, A325039, A350839, A352828, A352829, A352832.
%K nonn
%O 0,7
%A _Emeric Deutsch_, Apr 14 2006
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