

A183158


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


4



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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,4


LINKS

Table of n, a(n) for n=0..65.
R. Kehinde, A. Umar, On the semigroup of partial isometries of a finite chain, arXiv:1101.0049


FORMULA

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


EXAMPLE

T (4,2) = 6 because there are exactly 6 partial isometries (on a 4chain) 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 coordinatewise.
...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.


MAPLE

A183159 := proc(n) nh := floor(n/2) ; if type(n, 'even') then 13*4^nh12*nh^218*nh10; else 25*4^nh12*nh^230*nh22; 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 (* JeanFrançois Alcover, Nov 22 2017 *)


CROSSREFS

Cf. A183156 (row sums).
Sequence in context: A075397 A049429 A328647 * A174005 A152818 A302235
Adjacent sequences: A183155 A183156 A183157 * A183159 A183160 A183161


KEYWORD

nonn,tabl


AUTHOR

Abdullahi Umar, Dec 28 2010


STATUS

approved



