A118199 Number of partitions of n having no parts equal to the size of their Durfee squares.

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

Offset: 0

Author

Emeric Deutsch, Apr 14 2006

Keywords

• nonn

Comments

a(n) = A118198(n,0).

From Gus Wiseman, May 21 2022: (Start)

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/ a fixed point: A001522 (unproved), ranked by A352827 (cf. A352874).

Partitions w/o a fixed point: A064428 (unproved), ranked by A352826 (cf. A352873).

Partitions w/ a fixed point and a conjugate fixed point: A188674, reverse A325187, ranked by A353317.

Partitions w/o a fixed point or conjugate fixed point: A188674 (shifted).

(End)

Examples

			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.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Alois P. Heinz)

Crossrefs

Column k=0 of A118198.

A000041 counts partitions, strict A000009.

A000700 = self-conjugate partitions, ranked by A088902, complement A330644.

A002467 counts permutations with a fixed point, complement A000166.

A064410 counts partitions of crank 0, ranked by A342192.

A115720 and A115994 count partitions by Durfee square, rank stat A257990.

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.

Cf. A114088, A300788, A325039, A350839, A352828, A352829, A352832.

Programs

Maple

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))):
seq(a(n), n=0..70);  # Alois P. Heinz, Apr 09 2012

Mathematica

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 *)

Formula

G.f.: 1+sum(x^(k^2+k)/[(1-x^k)*product((1-x^i)^2, i=1..k-1)], k=1..infinity).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*n*sqrt(3)). - Vaclav Kotesovec, Jun 12 2025