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A238007
Number of strict partitions of n such that (greatest part) - (least part) >= (number of parts).
5
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
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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 17 2014
STATUS
approved