A056860 Triangle T(n,k) = number of element-subset partitions of {1..n} with n-k+1 equalities (n >= 1, 1 <= k <= n).

1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 12, 20, 15, 1, 5, 20, 50, 75, 52, 1, 6, 30, 100, 225, 312, 203, 1, 7, 42, 175, 525, 1092, 1421, 877, 1, 8, 56, 280, 1050, 2912, 5684, 7016, 4140, 1, 9, 72, 420, 1890, 6552, 17052, 31572, 37260, 21147

Offset: 1

Author

N. J. A. Sloane, Oct 13 2000

Keywords

• nonn
• tabl
• easy

Comments

T(n,k) = number of permutations on [n] with n in position k in which 321 patterns only occur as part of 3241 patterns. Example: T(4,2)=3 counts 1423, 2413, 3412. - David Callan, Jul 20 2005

From Gary W. Adamson, Feb 24 2011: (Start)

Given rows of an array such that n-th row is the eigensequence of an infinite lower triangular matrix with first n columns of Pascal's triangle and the rest zeros. The reoriented finite differences of the array starting from the top are the rows of A056860.

The first few rows of the array are

1, 1, 1, 1, 1, 1, ...

1, 2, 3, 4, 5, 6, ...

1, 2, 5, 10, 17, 26, ...

1, 2, 5, 15, 37, 76, ...

1, 2, 5, 15, 52, 151, ...

...

(End)

Examples

			T(n,k) starts:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  6,   5;
  1, 4, 12,  20,   15;
  1, 5, 20,  50,   75,   52;
  1, 6, 30, 100,  225,  312,   203;
  1, 7, 42, 175,  525, 1092,  1421,   877;
  1, 8, 56, 280, 1050, 2912,  5684,  7016,  4140;
  1, 9, 72, 420, 1890, 6552, 17052, 31572, 37260, 21147;
Building row sums Sum_{c=1..k} T(n,c), the following array results:
  1, 1,  1,   1,    1,    1,    1,     1,     1,     1, ...
  1, 2,  2,   2,    2,    2,    2,     2,     2,     2, ...
  1, 3,  5,   5,    5,    5,    5,     5,     5,     5, ...
  1, 4, 10,  15,   15,   15,   15,    15,    15,    15, ...
  1, 5, 17,  37,   52,   52,   52,    52,    52,    52, ...
  1, 6, 26,  76,  151,  203,  203,   203,   203,   203, ...
  1, 7, 37, 137,  362,  674,  877,   877,   877,   877, ...
  1, 8, 50, 225,  750, 1842, 3263,  4140,  4140,  4140, ...
  1, 9, 65, 345, 1395, 4307, 9991, 17007, 21147, 21147, ...
		

References

• W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000.

Links

• David Callan, A Combinatorial Interpretation of the Eigensequence for Composition, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.

Crossrefs

Essentially same as A056857, where rows are read from left to right.

T(2n+1,n+1) gives A124102.

T(2n,n) gives A297926.

Formula

T(n, k) = binomial(n-1, k-1)*B(k-1) where B denotes the Bell numbers A000110. - David Callan, Jul 20 2005

Extensions

More terms from David Callan, Jul 20 2005