A157046 -id:A157046 - cp's OEIS frontend

A157044 Triangle read by rows: T(n,k) = number of partitions of n into exactly k parts, each <= k.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 2, 1, 1, 0, 0, 2, 2, 1, 1, 0, 0, 2, 3, 2, 1, 1, 0, 0, 1, 4, 3, 2, 1, 1, 0, 0, 1, 4, 5, 3, 2, 1, 1, 0, 0, 0, 5, 6, 5, 3, 2, 1, 1, 0, 0, 0, 4, 8, 7, 5, 3, 2, 1, 1, 0, 0, 0, 4, 9, 10, 7, 5, 3, 2, 1, 1, 0, 0, 0, 3, 11, 12, 11, 7, 5, 3, 2, 1, 1, 0, 0, 0, 2, 11, 16, 14, 11, 7, 5

Offset: 1

Wouter Meeussen, Feb 22 2009

Also equals the number of partitions of n-1 into exactly k-1 parts, each <= k+1; equality reported by Franklin T. Adams-Watters.
Comment from N. J. A. Sloane: This is a special case of the fact that the number of partitions of a-c into exactly b-1 parts none exceeding c equals the number of partitions of a-b into exactly c-1 parts none exceeding b. See Andrews reference.
Row sums equal A064174; read backwards, rows approach the partition numbers A000041.

			Table begins:
1
0,1
0,1,1
0,1,1,1
0,0,2,1,1
0,0,2,2,1,1
0,0,2,3,2,1,1
0,0,1,4,3,2,1,1
0,0,1,4,5,3,2,1,1
0,0,0,5,6,5,3,2,1,1
0,0,0,4,8,7,5,3,2,1,1
0,0,0,4,9,10,7,5,3,2,1,1
0,0,0,3,11,12,11,7,5,3,2,1,1
0,0,0,2,11,16,14,11,7,5,3,2,1,1
0,0,0,1,12,19,19,15,11,7,5,3,2,1,1
0,0,0,1,11,23,24,21,15,11,7,5,3,2,1,1
0,0,0,0,11,25,31,27,22,15,11,7,5,3,2,1,1
0,0,0,0,9,29,37,36,29,22,15,11,7,5,3,2,1,1
0,0,0,0,8,30,46,45,39,30,22,15,11,7,5,3,2,1,1
0,0,0,0,6,32,52,58,50,41,30,22,15,11,7,5,3,2,1,1
0,0,0,0,5,32,61,70,66,53,42,30,22,15,11,7,5,3,2,1,1
0,0,0,0,3,32,68,86,82,71,55,42,30,22,15,11,7,5,3,2,1,1
0,0,0,0,2,30,76,101,104,90,74,56,42,30,22,15,11,7,5,3,2,1,1
0,0,0,0,1,29,81,120,127,116,95,76,56,42,30,22,15,11,7,5,3,2,1,1
a(8,4)=4 since {2,2,2,2}, {3,2,2,1}, {3,3,1,1}, {4,2,1,1} are the partitions of 8 in exactly 4 parts <=4.

George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976 (Theorem 1.5).

Cf. A000041, A157045, A157046, A047993

Table[T[n-1,k-1,k+1]-T[n-1,k-2,k+1],{n,20},{k,n}] with T[n,a,b] as defined in A047993.
<

A157045 Triangular table: number of partitions of n into exactly n-k parts, each <= n-k. Same as A157044 but with rows reversed.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 0, 1, 1, 2, 2, 0, 0, 1, 1, 2, 3, 2, 0, 0, 1, 1, 2, 3, 4, 1, 0, 0, 1, 1, 2, 3, 5, 4, 1, 0, 0, 1, 1, 2, 3, 5, 6, 5, 0, 0, 0, 1, 1, 2, 3, 5, 7, 8, 4, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 9, 4, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 12, 11, 3, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 14, 16, 11

Offset: 1

Wouter Meeussen, Feb 22 2009

See A157044. Rows approach the partition numbers.

George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976 (Theorem 1.5).

Cf. A000041, A157044, A157046, A047993

Table[T[n-1,n-k,n-k+2]-T[n-1,n-k-1,n-k+2],{n,1,9},{k,1,n}] with T[n,a,b] as defined in A047993.

cp's OEIS Frontend

A157044 Triangle read by rows: T(n,k) = number of partitions of n into exactly k parts, each <= k.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica