A183154 - OEIS

%I #16 Jan 17 2018 09:00:05

%S 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,

%T 7,21,35,35,21,7,1,495,8,28,56,70,56,28,8,1,1005,9,36,84,126,126,84,

%U 36,9,1,2027,10,45,120,210,252,210,120,45,10,1

%N 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)

%H R. Kehinde and A. Umar, <a href="http://arxiv.org/abs/1101.2558">On the semigroup of partial isometries of a finite chain</a>, arXiv:1101.2558 [math.GR], 2011.

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

%e 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.

%e Triangle starts as:

%e 1;

%e 1, 1;

%e 3, 2, 1;

%e 9, 3, 3, 1;

%e 23, 4, 6, 4, 1;

%e 53, 5, 10, 10, 5, 1;

%e 115, 6, 15, 20, 15, 6, 1;

%p A183155 := proc(n) 2^(n+1)-2*n-1 ; end proc:

%p A183154 := proc(n,k) if k =0 then A183155(n); else binomial(n,k) ; end if; end proc: # _R. J. Mathar_, Jan 06 2011

%t T[n_, k_] := If[k == 0, 2^(n + 1) - 2n - 1, Binomial[n, k]];

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 17 2018 *)

%o (PARI) A183155(n)=2^(n+1) - (2*n+1);

%o T(n,k)=if(k==0, A183155(n), binomial(n,k));

%o for(n=0,17,for(k=0,n,print1(T(n,k),", "));print()) \\ _Joerg Arndt_, Dec 30 2010

%Y Cf. A007318, A097813, A183155.

%K nonn,easy,tabl

%O 0,4

%A _Abdullahi Umar_, Dec 28 2010