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

Author

Wouter Meeussen, Feb 22 2009

Keywords

• nonn
• tabl

Comments

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.

Examples

			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.
		

References

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

Crossrefs

Cf. A000041, A157045, A157046, A047993

Programs

Mathematica

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.
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