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A240302 Number of partitions of n such that (maximal multiplicity of parts) > (multiplicity of the maximal part).

%I A240302 #12 Jun 05 2021 08:43:07
%S A240302 0,0,0,0,1,2,3,7,10,16,23,35,47,70,93,126,169,228,294,391,501,648,827,
%T A240302 1057,1329,1683,2105,2631,3266,4056,4992,6156,7538,9221,11234,13664,
%U A240302 16549,20033,24152,29077,34904,41844,50012,59710,71100,84541,100318,118869
%N A240302 Number of partitions of n such that (maximal multiplicity of parts) > (multiplicity of the maximal part).
%H A240302 Alois P. Heinz, <a href="/A240302/b240302.txt">Table of n, a(n) for n = 0..1000</a>
%F A240302 a(n) + A171979(n) = A000041(n) for n >= 1.
%e A240302 a(7) counts these 7 partitions: 511, 4111, 322, 3211, 31111, 22111, 211111.
%p A240302 b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
%p A240302       `if`(i<1, 0, b(n, i-1, k) +add(b(n-i*j, i-1, `if`(k=-1, j,
%p A240302       `if`(k=0, 0, `if`(j>k, 0, k)))), j=1..n/i)))
%p A240302     end:
%p A240302 a:= n-> b(n$2, -1):
%p A240302 seq(a(n), n=0..70);  # _Alois P. Heinz_, Apr 12 2014
%t A240302 z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]] (* maximal multiplicity *)
%t A240302 Table[Count[f[n], p_ /; m[p] == Count[p, Max[p]]], {n, 0, z}] (* A171979 *)
%t A240302 Table[Count[f[n], p_ /; m[p] > Count[p, Max[p]]], {n, 0, z}] (* A240302 *)
%t A240302 (* Second program: *)
%t A240302 b[n_, i_, k_] := b[n, i, k] = If[n == 0, If[k == 0, 1, 0],
%t A240302      If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i*j, i - 1, If[k == -1, j,
%t A240302      If[k == 0, 0, If[j > k, 0, k]]]], {j, 1, n/i}]]];
%t A240302 a[n_] := b[n, n, -1];
%t A240302 a /@ Range[0, 70] (* _Jean-François Alcover_, Jun 05 2021, after _Alois P. Heinz_ *)
%Y A240302 Cf. A240221, A000041.
%K A240302 nonn,easy
%O A240302 0,6
%A A240302 _Clark Kimberling_, Apr 04 2014