cp's OEIS Frontend

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).

%I A183154 #18 Jun 02 2025 03:15:07
%S A183154 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 A183154 7,21,35,35,21,7,1,495,8,28,56,70,56,28,8,1,1005,9,36,84,126,126,84,
%U A183154 36,9,1,2027,10,45,120,210,252,210,120,45,10,1
%N 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).
%H A183154 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 A183154 T(n,0) = A183155(n) and T(n,k) = binomial(n,k) if k > 0.
%e A183154 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 A183154 Triangle starts as:
%e A183154 1;
%e A183154 1, 1;
%e A183154 3, 2, 1;
%e A183154 9, 3, 3, 1;
%e A183154 23, 4, 6, 4, 1;
%e A183154 53, 5, 10, 10, 5, 1;
%e A183154 115, 6, 15, 20, 15, 6, 1;
%p A183154 A183155 := proc(n) 2^(n+1)-2*n-1 ; end proc:
%p A183154 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 A183154 T[n_, k_] := If[k == 0, 2^(n + 1) - 2n - 1, Binomial[n, k]];
%t A183154 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 17 2018 *)
%o A183154 (PARI) A183155(n)=2^(n+1) - (2*n+1);
%o A183154 T(n,k)=if(k==0, A183155(n), binomial(n,k));
%o A183154 for(n=0,17,for(k=0,n,print1(T(n,k),", "));print()) \\ _Joerg Arndt_, Dec 30 2010
%Y A183154 Cf. A007318, A097813, A183155.
%K A183154 nonn,easy,tabl
%O A183154 0,4
%A A183154 _Abdullahi Umar_, Dec 28 2010