A183153 T(n,k) is the number of order-preserving partial isometries of an n-chain of height k (height of alpha = |Im(alpha)|).

1, 1, 1, 1, 4, 1, 1, 9, 5, 1, 1, 16, 14, 6, 1, 1, 25, 30, 20, 7, 1, 1, 36, 55, 50, 27, 8, 1, 1, 49, 91, 105, 77, 35, 9, 1, 1, 64, 140, 196, 182, 112, 44, 10, 1, 1, 81, 204, 336, 378, 294, 156, 54, 11, 1, 1, 100, 285, 540, 714, 672, 450, 210, 65, 12, 1, 1, 121, 385, 825, 1254, 1386, 1122, 660

Offset: 0

Author

Abdullahi Umar, Dec 27 2010

Keywords

• nonn
• tabl
• easy

Comments

The matrix inverse starts

1;

-1,1;

3,-4,1;

-7,11,-5,1;

15,-26,16,-6,1;

-31,57,-42,22,-7,1;

63,-120,99,-64,29,-8,1;

-127,247,-219,163,-93,37,-9,1;

255,-502,466,-382,256,-130,46,-10,1;

...perhaps related to A054143. - R. J. Mathar, Mar 29 2013

Examples

			T(3,2)=5 because there are exactly 5 order-preserving partial isometries (on a 3-chain) of height 2, namely: (1,2)-->(1,2); (1,2)-->(2,3); (2,3)-->(1,2); (2,3)-->(2,3); (1,3)-->(1,3), the mappings are coordinate-wise.
Triangle begins as:
1;
1, 1;
1, 4, 1;
1, 9, 5, 1;
1, 16, 14, 6, 1;
1, 25, 30, 20, 7, 1;
1, 36, 55, 50, 27, 8, 1;
1, 49, 91, 105, 77, 35, 9, 1;
		

Links

• R. Kehinde, A. Umar, On the semigroup of partial isometries of a finite chain, arXiv:1101.0049

Crossrefs

Cf. A097813 (row sums), A125165, A121306, A029653.

Programs

PARI

T(n,k)=if(k==0,1, (2*n-k+1)*binomial(n,k)/(k+1));
for(n=0,17,for(k=0,n,print1(T(n,k),", ")))

Formula

T(n,0)=1. T(n,k)=(2*n-k+1)*C(n,k)/(k+1) if k>0.