A241818 - OEIS

%I #5 May 06 2014 15:06:23

%S 1,1,2,3,4,5,7,9,12,17,20,27,37,46,59,81,102,130,170,212,273,344,432,

%T 539,679,843,1049,1297,1602,1968,2422,2961,3608,4395,5334,6467,7800,

%U 9418,11311,13593,16287,19482,23214,27702,32908,39117,46305,54856,64749

%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.

%C For the partition [n] of n, "max(x(i) - x(i-1))" is (as in the Mathematica program) interpreted as 0.

%F a(n) + A241820(n) + A241821(n) = A000041(n) for n >= 0.

%e a(6) counts these 7 partitions:  6, 33, 321, 222, 2211, 21111, 111111.

%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. A241819, A241820, A241821, A241822, A000041.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Apr 30 2014