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

%I A100883 #20 Jan 22 2019 07:45:22
%S A100883 1,1,2,3,5,6,11,13,19,26,36,43,64,77,102,129,169,205,268,323,413,504,
%T A100883 629,751,947,1131,1384,1661,2024,2393,2919,3442,4136,4884,5834,6836,
%U A100883 8162,9531,11262,13155,15493,17981,21138,24472,28571,33066,38475,44305
%N A100883 Number of partitions of n in which the sequence of frequencies of the summands is nondecreasing.
%C A100883 From _Gus Wiseman_, Jan 21 2019: (Start)
%C A100883 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:
%C A100883   8   44   2222   11111111
%C A100883 .
%C A100883   1   2   11   3   111   22   1111   11   11111   1111   111111
%C A100883   7   6   6    5   5     4    4      33   3       22     2
%C A100883 .
%C A100883   1   1   11   111
%C A100883   2   3   2    2
%C A100883   5   4   4    3
%C A100883 (End)
%H A100883 Alois P. Heinz, <a href="/A100883/b100883.txt">Table of n, a(n) for n = 0..1000</a>
%e A100883 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.
%p A100883 b:= proc(n, i, t) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p A100883       `if`(i=1, `if`(n>=t, 1, 0), `if`(i=0, 0, b(n, i-1, t)+
%p A100883        add(b(n-i*j, i-1, j), j=t..floor(n/i))))))
%p A100883     end:
%p A100883 a:= n-> b(n$2, 1):
%p A100883 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jul 03 2014
%t A100883 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_ *)
%t A100883 Table[Length[Select[IntegerPartitions[n],OrderedQ[Length/@Split[#]]&]],{n,20}] (* _Gus Wiseman_, Jan 21 2019 *)
%Y A100883 Cf. A100881, A100882, A100884.
%Y A100883 Cf. A000085, A000219, A003293, A006951, A100471, A323582.
%K A100883 nonn
%O A100883 0,3
%A A100883 _David S. Newman_, Nov 21 2004
%E A100883 More terms from _Vladeta Jovovic_, Nov 23 2004