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Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >= ... >= x(k), of n such that max(x(i) - x(i-1)) = number of distinct parts of p.
6

%I #4 May 06 2014 15:06:35

%S 0,0,0,0,1,1,2,2,5,3,10,10,13,18,25,25,39,48,54,78,95,113,142,183,215,

%T 270,322,396,480,587,686,845,1022,1210,1453,1730,2081,2459,2945,3454,

%U 4108,4838,5744,6707,7959,9216,10938,12692,14934,17346,20296,23526

%N Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >= ... >= x(k), of n such that max(x(i) - x(i-1)) = number of distinct parts of p.

%F a(n) = A241819(n) - A241818(n) for n >= 0.

%F a(n) + A241818(n) + A241822 (n) = A000041(n) for n >= 0.

%e a(6) counts these 2 partitions: 42, 3111.

%t z = 30; f[n_] := f[n] = IntegerPartitions[n]; d[p_] := d[p] = Length[DeleteDuplicates[p]]; g[p_] := Max[-Differences[p]];

%t Table[Count[f[n], p_ /; g[p] < d[p]], {n, 0, z}] (* A241818 *)

%t Table[Count[f[n], p_ /; g[p] <= d[p]], {n, 0, z}] (* A241819 *)

%t Table[Count[f[n], p_ /; g[p] == d[p]], {n, 0, z}] (* A241820 *)

%t Table[Count[f[n], p_ /; g[p] >= d[p]], {n, 0, z}] (* A241821 *)

%t Table[Count[f[n], p_ /; g[p] > d[p]], {n, 0, z}] (* A241822 *)

%Y Cf. A241818, A241819, A241821, A241822, A000041.

%K nonn,easy

%O 0,7

%A _Clark Kimberling_, Apr 30 2014