A238007 Number of strict partitions of n such that (greatest part) - (least part) >= (number of parts).

0, 0, 0, 1, 1, 2, 3, 5, 5, 8, 10, 13, 16, 20, 23, 31, 36, 43, 52, 62, 72, 87, 102, 120, 139, 163, 188, 220, 254, 292, 338, 389, 444, 510, 581, 665, 758, 862, 978, 1111, 1258, 1422, 1608, 1814, 2042, 2302, 2588, 2908, 3261, 3655, 4093, 4580, 5118, 5714, 6374

Offset

1,6

Comments

From Omar E. Pol, Mar 04 2017: (Start)

Partitions into distinct parts are sometimes called "strict partitions".

a(n) is also the number of partitions of n into distinct parts, which are not the partitions into (one or more) consecutive parts. (End)

Links

Robert Israel, Table of n, a(n) for n = 1..200

Formula

a(n) = A000009(n) - A001227(n). - Omar E. Pol, Mar 04 2017

a(n) = A238005(n)+A238006(n). - R. J. Mathar, Sep 08 2021

Example

a(9) = 5 counts these partitions:  81, 72, 63, 621, 531.

Maple

spart:= proc(n, a, b, k) option remember;
# count strict partitions of n in exactly k parts with parts in [a, b]
if min(k, n) = 0 then if n=k then return 1 else return 0 fi fi;
if n < k*(2*a+k-1)/2 or n > k*(2*b-k+1)/2 then return 0 fi;
add (procname(n-x, a, x-1, k-1), x=a..min(n, b));
end proc:
f:= n -> add(add(add(spart(n-a-b, a+1, b-1, k-2), k=2..b-a), b=a+2..n), a=1..n-2):
map(f, [$1..100]); # Robert Israel, Mar 06 2017

Mathematica

z = 70; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]];
Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A001227 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A003056 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A238005 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A238006 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A238007 *)

Crossrefs

Cf. A000009, A001227, A003056, A238005, A238006.

Sequence in context: A237825 A194939 A135635 * A204207 A354704 A287919

Adjacent sequences: A238004 A238005 A238006 * A238008 A238009 A238010

Keyword

nonn,easy

Author

Clark Kimberling, Feb 17 2014

Status

approved