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A100883
Number of partitions of n in which the sequence of frequencies of the summands is nondecreasing.
49
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
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 *)
KEYWORD
nonn
AUTHOR
David S. Newman, Nov 21 2004
EXTENSIONS
More terms from Vladeta Jovovic, Nov 23 2004
STATUS
approved