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)|).

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, 28, 56, 70, 56, 28, 8, 1, 1, 36, 84, 126, 126, 84, 36, 9, 1, 1, 45, 120, 210, 252, 210, 120, 45, 10, 1, 1, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 1, 66, 220

Offset: 0

Author

Abdullahi Umar, Jan 12 2011

Keywords

• nonn
• tabl
• easy

Comments

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

Examples

			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);
    (1,4)-->(1,4) - the mappings are coordinate-wise
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, 28, 56, 70
		

Links

• R. Kehinde, S. O. Makanjuola and A. Umar, On the semigroup of order-decreasing partial isometries of a finite chain, arXiv:1101.2558 [math.GR], 2011.

Crossrefs

Cf. A007318; Row sums are A000325 for n >= 0.

Programs

Mathematica

z = 14; p[n_, x_] := (x + 1) p[n - 1, x] + (n - 1)*x; p[0, x_] = 1;
t = Table[Factor[p[n, x]], {n, 0, z}]
TableForm[Rest[Table[CoefficientList[t[[n]], x], {n, 0, z}]]] (* A184049 array *)
Flatten[CoefficientList[t, x]] (* A184049 sequence *)
(* Clark Kimberling, Dec 02 2014 *)

Formula

T(n;0)=1 and T(n,k)=C(n+1,k+1), (k>0)

Extensions

More terms from Clark Kimberling, Dec 02 2014