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

Author

Clark Kimberling, Feb 17 2014

Keywords

• nonn
• easy

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)

Examples

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

Links

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

Crossrefs

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

Programs

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 *)

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