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

0,3

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)

Links

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

Example

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.

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 *)

Crossrefs

Cf. A100881, A100882, A100884.

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

Sequence in context: A091909 A171040 A317256 * A240303 A240218 A347414

Adjacent sequences: A100880 A100881 A100882 * A100884 A100885 A100886

Keyword

nonn

Author

David S. Newman, Nov 21 2004

Extensions

More terms from Vladeta Jovovic, Nov 23 2004

Status

approved