cp's OEIS Frontend

A183153 T(n,k) is the number of order-preserving partial isometries of an n-chain of height k (height of alpha = |Im(alpha)|).

%I A183153 #15 Mar 29 2013 19:04:31
%S A183153 1,1,1,1,4,1,1,9,5,1,1,16,14,6,1,1,25,30,20,7,1,1,36,55,50,27,8,1,1,
%T A183153 49,91,105,77,35,9,1,1,64,140,196,182,112,44,10,1,1,81,204,336,378,
%U A183153 294,156,54,11,1,1,100,285,540,714,672,450,210,65,12,1,1,121,385,825,1254,1386,1122,660
%N A183153 T(n,k) is the number of order-preserving partial isometries of an n-chain of height k (height of alpha = |Im(alpha)|).
%C A183153 The matrix inverse starts
%C A183153 1;
%C A183153 -1,1;
%C A183153 3,-4,1;
%C A183153 -7,11,-5,1;
%C A183153 15,-26,16,-6,1;
%C A183153 -31,57,-42,22,-7,1;
%C A183153 63,-120,99,-64,29,-8,1;
%C A183153 -127,247,-219,163,-93,37,-9,1;
%C A183153 255,-502,466,-382,256,-130,46,-10,1;
%C A183153 ...perhaps related to A054143. - _R. J. Mathar_, Mar 29 2013
%H A183153 R. Kehinde, A. Umar, <a href="http://arxiv.org/abs/1101.0049">On the semigroup of partial isometries of a finite chain</a>, arXiv:1101.0049
%F A183153 T(n,0)=1. T(n,k)=(2*n-k+1)*C(n,k)/(k+1) if k>0.
%e A183153 T(3,2)=5 because there are exactly 5 order-preserving partial isometries (on a 3-chain) of height 2, namely: (1,2)-->(1,2); (1,2)-->(2,3); (2,3)-->(1,2); (2,3)-->(2,3); (1,3)-->(1,3), the mappings are coordinate-wise.
%e A183153 Triangle begins as:
%e A183153 1;
%e A183153 1, 1;
%e A183153 1, 4, 1;
%e A183153 1, 9, 5, 1;
%e A183153 1, 16, 14, 6, 1;
%e A183153 1, 25, 30, 20, 7, 1;
%e A183153 1, 36, 55, 50, 27, 8, 1;
%e A183153 1, 49, 91, 105, 77, 35, 9, 1;
%o A183153 (PARI) T(n,k)=if(k==0,1, (2*n-k+1)*binomial(n,k)/(k+1));
%o A183153 for(n=0,17,for(k=0,n,print1(T(n,k),", ")))
%Y A183153 Cf. A097813 (row sums), A125165, A121306, A029653.
%K A183153 nonn,tabl,easy
%O A183153 0,5
%A A183153 _Abdullahi Umar_, Dec 27 2010