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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). 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 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.

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-Fran├žois 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

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Last modified April 18 07:59 EDT 2021. Contains 343084 sequences. (Running on oeis4.)