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

0,4

Links

Table of n, a(n) for n=0..65.

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

Formula

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

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A007318, A097813, A183155.

Sequence in context: A086963 A079749 A156647 * A193791 A160760 A152860

Adjacent sequences: A183151 A183152 A183153 * A183155 A183156 A183157

Keyword

nonn,easy,tabl

Author

Abdullahi Umar, Dec 28 2010

Status

approved