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

0,5

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

Links

Table of n, a(n) for n=0..68.

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.

Formula

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A256697 A133567 A271665 * A125230 A208334 A162430

Adjacent sequences: A184046 A184047 A184048 * A184050 A184051 A184052

Keyword

nonn,tabl,easy

Author

Abdullahi Umar, Jan 12 2011

Extensions

More terms from Clark Kimberling, Dec 02 2014

Status

approved