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A238395
Number of partitions of n that sorted in increasing order contain a part k in position k for some k.
24
0, 1, 1, 2, 4, 5, 8, 12, 18, 25, 34, 47, 65, 88, 118, 154, 203, 263, 343, 442, 568, 721, 914, 1149, 1445, 1807, 2255, 2800, 3468, 4270, 5250, 6425, 7855, 9566, 11635, 14103, 17068, 20584, 24784, 29754, 35670, 42653, 50934, 60688, 72212, 85742, 101662, 120293
OFFSET
0,4
COMMENTS
Note that considering partitions in standard decreasing order, we obtain A001522.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Giovanni Resta)
FORMULA
a(n) + A238394(n) = p(n) = A000041(n).
EXAMPLE
a(6) = 11 - 3 = 8, because of the 11 partitions of 6 only 3 do not contain a 1 in position 1, a 2 in position 2, or a 3 in position 3, namely (3,3), (2,4) and (6).
MAPLE
b:= proc(n, i) option remember; `if`(n=0, [0, 1],
`if`(i<1, [0$2], b(n, i-1) +`if`(i>n, 0,
(p->[p[1] +coeff(p[2], x, i-1), expand(x*(p[2]-
coeff(p[2], x, i-1)*x^(i-1)))])(b(n-i, i)))))
end:
a:= n-> b(n$2)[1]:
seq(a(n), n=0..70); # Alois P. Heinz, Feb 26 2014
MATHEMATICA
a[n_] := Length@ Select[IntegerPartitions@ n, MemberQ[ Reverse@# - Range@ Length@#, 0] &]; Array[a, 30]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n==0, {0, 1}, If[i<1, {0, 0}, b[n, i-1] + If[i>n, 0, Function[p, {p[[1]] + Coefficient[p[[2]], x, i-1], x*(p[[2]] - Coefficient[p[[2]], x, i-1]*x^(i-1))}][b[n-i, i]]]]]; a[n_] := b[n, n][[1]]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Giovanni Resta, Feb 26 2014
STATUS
approved