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
KEYWORD
nonn
AUTHOR
David S. Newman, Nov 21 2004
EXTENSIONS
More terms from Vladeta Jovovic, Nov 23 2004
STATUS
approved