cp's OEIS Frontend

A184049 T(n,k) is the number of order-preserving and order-decreasing partial isometries (of an n-chain) of height k (height of alpha = |Im(alpha)|).

%I A184049 #18 Dec 14 2014 21:18:23
%S A184049 1,1,1,1,3,1,1,6,4,1,1,10,10,5,1,1,15,20,15,6,1,1,21,35,35,21,7,1,1,
%T A184049 28,56,70,56,28,8,1,1,36,84,126,126,84,36,9,1,1,45,120,210,252,210,
%U A184049 120,45,10,1,1,55,165,330,462,462,330,165,55,11,1,1,66,220
%N A184049 T(n,k) is the number of order-preserving and order-decreasing partial isometries (of an n-chain) of height k (height of alpha = |Im(alpha)|).
%C A184049 Row n gives the coefficients of the polynomial p(n,x) = (x + 1)*p(n-1,x) + (n - 1)*x, where p(0,x) = 1. - _Clark Kimberling_, Dec 02 2014
%H A184049 R. Kehinde, S. O. Makanjuola and A. Umar, <a href="http://arxiv.org/abs/1101.2558">On the semigroup of order-decreasing partial isometries of a finite chain</a>, arXiv:1101.2558 [math.GR], 2011.
%F A184049 T(n;0)=1 and T(n,k)=C(n+1,k+1), (k>0)
%e A184049 T (4,2) = 10 because there are exactly 10 order-preserving and order-decreasing partial isometries (on a 4-chain) of height 2, namely: (1,2)-->(1,2); (2,3)-->(1,2); (2,3)-->(2,3); (3,4)-->(1,2); (3,4)-->(2,3); (3,4)-->(3,4); (1,3)-->(1,3); (2,4)-->(1,3); (2,4)-->(2,4);
%e A184049     (1,4)-->(1,4) - the mappings are coordinate-wise
%e A184049 1,
%e A184049 1, 1,
%e A184049 1, 3, 1,
%e A184049 1, 6, 4, 1,
%e A184049 1, 10, 10, 5, 1,
%e A184049 1, 15, 20, 15, 6, 1,
%e A184049 1, 21, 35, 35, 21, 7, 1,
%e A184049 1, 28, 56, 70
%t A184049 z = 14; p[n_, x_] := (x + 1) p[n - 1, x] + (n - 1)*x; p[0, x_] = 1;
%t A184049 t = Table[Factor[p[n, x]], {n, 0, z}]
%t A184049 TableForm[Rest[Table[CoefficientList[t[[n]], x], {n, 0, z}]]] (* A184049 array *)
%t A184049 Flatten[CoefficientList[t, x]] (* A184049 sequence *)
%t A184049 (* _Clark Kimberling_, Dec 02 2014 *)
%Y A184049 Cf. A007318; Row sums are A000325 for n >= 0.
%K A184049 nonn,tabl,easy
%O A184049 0,5
%A A184049 _Abdullahi Umar_, Jan 12 2011
%E A184049 More terms from _Clark Kimberling_, Dec 02 2014