A058398 Partition triangle A008284 read from right to left.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 3, 1, 1, 1, 2, 3, 4, 3, 1, 1, 1, 2, 3, 5, 5, 4, 1, 1, 1, 2, 3, 5, 6, 7, 4, 1, 1, 1, 2, 3, 5, 7, 9, 8, 5, 1, 1, 1, 2, 3, 5, 7, 10, 11, 10, 5, 1, 1, 1, 2, 3, 5, 7, 11, 13, 15, 12, 6, 1, 1, 1, 2, 3, 5, 7, 11, 14, 18, 18, 14, 6, 1, 1, 1, 2, 3, 5, 7, 11

Offset: 1

Author

Wolfdieter Lang, Dec 11 2000

Keywords

• nonn
• easy
• tabl
• nice

Comments

a(n,m) is the number of partitions of n with n-(m-1) parts or, equivalently, with greatest part n-(m-1).

The columns are the diagonals of triangle A008284. The diagonals are the columns of the partition array p(n,m), n >= 0, m >= 1, with p(n,m) the number of partitions of n in which every part is <= m; p(0,m) := 1. For n >= 1 this array is obtained from table A026820 read as lower triangular array with extension of the rows according to p(n,m)=A000041(n) for m>n.

Examples

			Lower triangular matrix:
  1;
  1,1;
  1,1,1;
  1,1,2,1;
  1,1,2,2,1;
  1,1,2,3,3,1;
  1,1,2,3,4,3,1;
  1,1,2,3,5,5,4,1;
  ...
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.
• M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 27.

Links

• Seiichi Manyama, Rows n = 1..140, flattened
• Henry Bottomley, Illustration of initial terms
• Roser Homs and Anna-Lena Winz, Deformations of local Artin rings via Hilbert-Burch matrices, arXiv:2309.06871 [math.AC], 2023. See p. 16.

Crossrefs

Cf. A026820, A008284, A000041.

Programs

Mathematica

row[n_] := Table[ IntegerPartitions[n, k] // Length, {k, 0, n}] // Differences // Reverse; Table[row[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Feb 28 2013 *)

Formula

a(n, m)= A008284(n, n-(m-1)).

a(n, m)= p(m-1, n-m+1), n >= m >= 1 with the p(n, m) array defined in the comment.

a(n, m)=0 if n

Viewed as a square array by antidiagonals, T(n,k) = 0 if n<0; T(n,1) = 1; otherwise T(n,k) = T(n,k-1) + T(n-k,k). - Franklin T. Adams-Watters, Jul 25 2006

Let x be a triangular number C(n,2), where n is the integer being partitioned. Then a(x) = a(x+1) = a(x+2) = 1. Also, a(x+3) = 2 for x>3 and a(x-1) = floor(n/2). - Allan Bickle, Apr 18 2024

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