A100883 Number of partitions of n in which the sequence of frequencies of the summands is nondecreasing.

1, 1, 2, 3, 5, 6, 11, 13, 19, 26, 36, 43, 64, 77, 102, 129, 169, 205, 268, 323, 413, 504, 629, 751, 947, 1131, 1384, 1661, 2024, 2393, 2919, 3442, 4136, 4884, 5834, 6836, 8162, 9531, 11262, 13155, 15493, 17981, 21138, 24472, 28571, 33066, 38475, 44305

Offset: 0

Author

David S. Newman, Nov 21 2004

Keywords

• nonn

Comments

From Gus Wiseman, Jan 21 2019: (Start)

Also the number of semistandard Young tableaux where the rows are constant and the entries sum to n. For example, the a(8) = 19 tableaux are:

8 44 2222 11111111

.

1 2 11 3 111 22 1111 11 11111 1111 111111

7 6 6 5 5 4 4 33 3 22 2

.

1 1 11 111

2 3 2 2

5 4 4 3

(End)

Examples

			a(5) = 6 because, of the 7 unrestricted partitions of 5, only one, 2 + 2 + 1, has a decreasing sequence of frequencies. Two is used twice, but 1 is used only once.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A100881, A100882, A100884.

Cf. A000085, A000219, A003293, A006951, A100471, A323582.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n<0, 0, `if`(n=0, 1,
      `if`(i=1, `if`(n>=t, 1, 0), `if`(i=0, 0, b(n, i-1, t)+
       add(b(n-i*j, i-1, j), j=t..floor(n/i))))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 03 2014

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n<0, 0, If[n == 0, 1, If[i == 1, If[n >= t, 1, 0], If[i == 0, 0, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, t, Floor[n/i]}]]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OrderedQ[Length/@Split[#]]&]],{n,20}] (* Gus Wiseman, Jan 21 2019 *)

Extensions

More terms from Vladeta Jovovic, Nov 23 2004