A183158 T(n,k) is the number of partial isometries of an n-chain of fix k (fix of alpha is the number of fixed points of alpha).

1, 1, 1, 4, 2, 1, 12, 6, 3, 1, 38, 10, 6, 4, 1, 90, 26, 10, 10, 5, 1, 220, 42, 15, 20, 15, 6, 1, 460, 106, 21, 35, 35, 21, 7, 1, 1018, 170, 28, 56, 70, 56, 28, 8, 1, 2022, 426, 36, 84, 126, 126, 84, 36, 9, 1, 4304, 682, 45, 120, 210, 252, 210, 120, 45, 10, 1

Offset: 0

Author

Abdullahi Umar, Dec 28 2010

Keywords

• nonn
• tabl

Examples

			T (4,2) = 6 because there are exactly 6 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.
...1.
...1....1.
...4....2....1.
..12....6....3....1.
..38...10....6....4....1.
..90...26...10...10....5....1.
.220...42...15...20...15....6....1.
.460..106...21...35...35...21....7....1.
1018..170...28...56...70...56...28....8....1.
2022..426...36...84..126..126...84...36....9....1.
4304..682...45..120..210..252..210..120...45...10....1.
		

Links

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

Crossrefs

Cf. A183156 (row sums).

Programs

Maple

A183159 := proc(n) nh := floor(n/2) ; if type(n,'even') then 13*4^nh-12*nh^2-18*nh-10; else 25*4^nh-12*nh^2-30*nh-22; end if; %/3 ; end proc:
A061547 := proc(n) 3*2^n/8 +(-2)^n/24 - 2/3; end proc:
A183158 := proc(n,k) if k = 0 then A183159(n) ; elif k = 1 then A061547(n+1) ; else binomial(n,k) ; end if; end proc: # R. J. Mathar, Jan 06 2011

Mathematica

T[n_, 0] := (51*2^n + (-2)^n - 40)/12 - n*(n + 3);
T[n_, 1] := (9*2^n + (-1)^(n+1)*2^n - 8)/12;
T[n_, k_] := Binomial[n, k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 22 2017 *)

Formula

T(n,0)= A183159(n). T(n,1)=A061547(n+1). T(n,k)=binomial(n,k), k > 1.