A118053 -id:A118053 - cp's OEIS frontend

A240313 Number of partitions p of n such that (maximal multiplicity of the parts of p) >= (maximal part of p).

1, 1, 1, 1, 3, 3, 5, 5, 8, 11, 15, 19, 27, 32, 43, 53, 70, 84, 112, 135, 174, 212, 268, 324, 407, 490, 606, 731, 897, 1075, 1312, 1567, 1899, 2265, 2726, 3238, 3886, 4598, 5486, 6482, 7698, 9063, 10727, 12592, 14846, 17391, 20427, 23862, 27952, 32568, 38033

Offset: 0

Clark Kimberling, Apr 05 2014

			a(6) counts these 5 partitions:  3111, 222, 2211, 21111, 111111.

Cf. A240310, A240311, A240312, A240314, A000041, A118053.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *)
Table[Count[f[n], p_ /; m[p] < Max[p]], {n, 0, z}]  (* A240310 *)
Table[Count[f[n], p_ /; m[p] <= Max[p]], {n, 0, z}] (* A240311 *)
Table[Count[f[n], p_ /; m[p] == Max[p]], {n, 0, z}] (* A240312 *)
Table[Count[f[n], p_ /; m[p] >= Max[p]], {n, 0, z}] (* A240313 *)
Table[Count[f[n], p_ /; m[p] > Max[p]], {n, 0, z}]  (* A240314 *)

a(n) = A240312(n) + A240314(n) for n >= 0.

a(n) + A240310(n) = A000041(n) for n >= 0.

A118052 Number of partitions of n which contain their signature as a subpartition.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 8, 10, 16, 22, 32, 42, 58, 75, 101, 131, 174, 223, 293, 372, 480, 607, 772, 968, 1220, 1517, 1895, 2345, 2908, 3576, 4408, 5390, 6604, 8038, 9788, 11853, 14366, 17315, 20881, 25070, 30098, 35990, 43034, 51272, 61074, 72522, 86073, 101878, 120520

Offset: 0

Franklin T. Adams-Watters, Apr 10 2006

nonn

What is lim_{n->infinity} a(n)/p(n) (where p(n) = A000041(n) is the partition function)? It appears to be converging to something close to 0.8.

The limit must be at least 0.83846 = a(64)/p(64) and is probably closer to 0.9. - Charlie Neder, Aug 30 2018

			For n=3, signature([3]) = [1] is a subpartition of [3], signature([2,1]) = [1^2] is a subpartition of [2,1], but signature([1^3]) = [3] is not a subpartition of [1^3], so a(3)=2.

Charlie Neder, Table of n, a(n) for n = 0..64

Cf. A000041, A115621, A115622, A000041, A118053, A118054.

a(26) onwards from Charlie Neder, Aug 30 2018

cp's OEIS Frontend

A240313 Number of partitions p of n such that (maximal multiplicity of the parts of p) >= (maximal part of p).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A118052 Number of partitions of n which contain their signature as a subpartition.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions