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
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
KEYWORD
AUTHOR
Abdullahi Umar, Jan 12 2011
EXTENSIONS
More terms from Clark Kimberling, Dec 02 2014
STATUS
approved