A209354 Triangular array: T(n,k) = number of partitions of n for which (maximal term)-(minimal term)=k, if 0<=k<n, and T(n,n)=1.

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 3, 1, 1, 0, 1, 3, 2, 3, 1, 1, 0, 1, 1, 5, 3, 3, 1, 1, 0, 1, 3, 4, 6, 3, 3, 1, 1, 0, 1, 2, 6, 6, 7, 3, 3, 1, 1, 0, 1, 3, 6, 10, 7, 7, 3, 3, 1, 1, 0, 1, 1, 9, 10, 12, 8, 7, 3, 3, 1, 1, 0, 1, 5, 6, 15, 14, 13, 8, 7, 3, 3, 1, 1, 0, 1, 1, 11, 15, 20

Offset

1,11

Comments

Row sums: A000041 (number of partitions of n).

Links

Table of n, a(n) for n=1..95.

Example

First ten rows:
  1
  0 1
  1 0 1
  1 1 0 1
  2 1 1 0 1
  1 3 1 1 0 1
  3 2 3 1 1 0 1
  1 5 3 3 1 1 0 1
  3 4 6 3 3 1 1 0 1
  2 6 6 7 3 3 1 1 0 1
Row 5 (counting the top row as row 0):
  T(5,0)=1 counts [1,1,1,1,1]
  T(5,1)=3 counts [2,1,1,1], [2,2,1], [3,2]
  T(5,2)=1 counts [3,1,1]
  T(5,3)=1 counts [4,1]
  T(5,4)=0
  T(5,5)=1 counts [5]

Mathematica

f[n_] := IntegerPartitions[n];
p[n_, k_] := f[n][[k]];
r[n_] := Table[p[n, k], {k, 1, Length[f[n]]}]
g[n_, k_] := Max[p[n, k]] - Min[p[n, k]]; g[n_, 1] := n;
t[n_] := Table[g[n, k], {k, 1, Length[f[n]]}]
c[0, 0] = 1; c[n_, k_] := Count[t[n], k]
u = Table[c[n, k], {n, 0, 15}, {k, 0, n}];
TableForm[u] (* as a triangle *)
Flatten[u]   (* as a sequence *)

Crossrefs

Cf. A000041.

Sequence in context: A328610 A217605 A096651 * A294446 A318163 A114640

Adjacent sequences: A209351 A209352 A209353 * A209355 A209356 A209357

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 06 2012

Status

approved