login
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 15:06:51

%S 0,0,0,0,1,2,4,6,10,13,22,29,40,55,76,95,129,167,215,278,354,448,570,

%T 716,896,1115,1387,1713,2116,2597,3182,3881,4741,5748,6976,8416,10177,

%U 12219,14704,17592,21051,25101,29960,35559,42267,50017,59253,69898,82524

%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) = A241820(n) + A241822(n) for n >= 0.

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

%e a(6) counts these 4 partitions: 51, 42, 411, 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, A241820, A241822, A000041.

%K nonn,easy

%O 0,6

%A _Clark Kimberling_, Apr 30 2014