A183154 T(n,k) is the number of order-preserving partial isometries (of an n-chain) of fixed k (fix of alpha is the number of fixed points of alpha).

1, 1, 1, 3, 2, 1, 9, 3, 3, 1, 23, 4, 6, 4, 1, 53, 5, 10, 10, 5, 1, 115, 6, 15, 20, 15, 6, 1, 241, 7, 21, 35, 35, 21, 7, 1, 495, 8, 28, 56, 70, 56, 28, 8, 1, 1005, 9, 36, 84, 126, 126, 84, 36, 9, 1, 2027, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1

Offset: 0

Author

Abdullahi Umar, Dec 28 2010

Keywords

• nonn
• easy
• tabl

Examples

			T (4,2) = 6 because there are exactly 6 order-preserving partial isometries (on a 4-chain) of fix 2, namely: (1,2)-->(1,2); (2,3)-->(2,3); (3,4)-->(3,4); (1,3)-->(1,3); (2,4)-->(2,4); (1,4)-->(1,4) - the mappings are coordinate-wise.
Triangle starts as:
1;
1, 1;
3, 2, 1;
9, 3, 3, 1;
23, 4, 6, 4, 1;
53, 5, 10, 10, 5, 1;
115, 6, 15, 20, 15, 6, 1;
		

Links

• R. Kehinde and A. Umar, On the semigroup of partial isometries of a finite chain, arXiv:1101.2558 [math.GR], 2011.

Crossrefs

Cf. A007318, A097813, A183155.

Programs

Maple

A183155 := proc(n) 2^(n+1)-2*n-1 ; end proc:
A183154 := proc(n,k) if k =0 then A183155(n); else binomial(n,k) ; end if; end proc: # R. J. Mathar, Jan 06 2011

Mathematica

T[n_, k_] := If[k == 0, 2^(n + 1) - 2n - 1, Binomial[n, k]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 17 2018 *)

PARI

A183155(n)=2^(n+1) - (2*n+1);
T(n,k)=if(k==0, A183155(n), binomial(n,k));
for(n=0,17,for(k=0,n,print1(T(n,k),", "));print()) \\ Joerg Arndt, Dec 30 2010

Formula

T(n,0) = A183155(n) and T(n,k) = binomial(n,k) if k > 0.