cp's OEIS Frontend

A239950 Number of partitions of n such that (number of distinct parts) = least part.

%I A239950 #18 Nov 17 2015 01:41:25
%S A239950 0,1,1,1,1,2,2,3,4,4,6,6,9,8,14,11,19,18,25,24,37,31,50,46,61,64,86,
%T A239950 79,112,115,136,149,190,184,239,255,293,329,382,408,489,531,595,675,
%U A239950 772,827,952,1066,1176,1320,1468,1627,1827,2030,2219,2493,2769,3053
%N A239950 Number of partitions of n such that (number of distinct parts) = least part.
%C A239950 Also for n>0 the number of partitions of n such that (number of distinct parts) = multiplicity of the greatest part (by conjugation of the partition table). - _Joerg Arndt_, Apr 28 2014
%H A239950 Alois P. Heinz, <a href="/A239950/b239950.txt">Table of n, a(n) for n = 0..1000</a>
%F A239950 A239948(n) + a(n) + A239951(n) = A000041(n) for n >= 0.
%e A239950 a(8) counts these 4 partitions :  62, 422, 332, 11111111.
%p A239950 b:= proc(n, i, d) option remember; `if`(min(i, n)<d+1, 0,
%p A239950       `if`(irem(n, i)=0 and i=d+1, 1, b(n, i-1, d)+
%p A239950       add(b(n-i*j, i-1, d+1), j=1..n/i)))
%p A239950     end:
%p A239950 a:= n-> b(n$2, 0):
%p A239950 seq(a(n), n=0..60);  # _Alois P. Heinz_, Apr 02 2014
%t A239950 z = 50; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[n_] := f[n] = IntegerPartitions[n];
%t A239950 Table[Count[f[n], p_ /; d[p] < Min[p]], {n, 0, z}]  (*A239948*)
%t A239950 Table[Count[f[n], p_ /; d[p] <= Min[p]], {n, 0, z}] (*A239949*)
%t A239950 Table[Count[f[n], p_ /; d[p] == Min[p]], {n, 0, z}] (*A239950*)
%t A239950 Table[Count[f[n], p_ /; d[p] > Min[p]], {n, 0, z}]  (*A239951*)
%t A239950 Table[Count[f[n], p_ /; d[p] >= Min[p]], {n, 0, z}] (*A239952*)
%t A239950 b[n_, i_, d_] := b[n, i, d] = If[Min[i, n]<d+1, 0, If[Mod[n, i]==0 && i == d+1, 1, b[n, i-1, d] + Sum[b[n-i*j, i-1, d+1], {j, 1, n/i}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Nov 17 2015, after _Alois P. Heinz_ *)
%Y A239950 Cf. A239948, A239949, A239951, A239952.
%K A239950 nonn,easy
%O A239950 0,6
%A A239950 _Clark Kimberling_, Mar 30 2014