A183157 Triangle read by rows: T(n,k) is the number of partial isometries of an n-chain of height k (height of alpha = |Im(alpha)|).

1, 1, 1, 1, 4, 2, 1, 9, 10, 2, 1, 16, 28, 12, 2, 1, 25, 60, 40, 14, 2, 1, 36, 110, 100, 54, 16, 2, 1, 49, 182, 210, 154, 70, 18, 2, 1, 64, 280, 392, 364, 224, 88, 20, 2, 1, 81, 408, 672, 756, 588, 312, 108, 22, 2, 1, 100, 570, 1080, 1428, 1344, 900, 420, 130, 24, 2

Offset

0,5

Comments

Rows also give the coefficients of the clique polynomial of the n X n bishop graph. - Eric W. Weisstein, Jun 04 2017

Links

Table of n, a(n) for n=0..65.

R. Kehinde and A. Umar, On the semigroup of partial isometries of a finite chain, arXiv:1101.2558 [math.GR], 2011.

Eric Weisstein's World of Mathematics, Bishop Graph

Eric Weisstein's World of Mathematics, Clique Polynomial

Formula

T(n,0)=1, T(n,1) = n^2 and T(n,k)=2*(2*n-k+1)*binomial(n,k)/(k+1), k > 1.

Example

T (3,2) = 10 because there are exactly 10 partial isometries (on a 3-chain) of height 2, namely: (1,2)-->(1,2); (1,2)-->(2,1); (1,2)-->(2,3); (1,2)-->(3,2); (2,3)-->(1,2); (2,3)-->(2,1); (2,3)-->(2,3); (2,3)-->(3,2); (1,3)-->(1,3); (1,3)-->(3,1) - the mappings are coordinate-wise.
The triangle starts
  1;
  1,    1;
  1,    4,    2;
  1,    9,   10,    2;
  1,   16,   28,   12,    2;
  1,   25,   60,   40,   14,    2;
  1,   36,  110,  100,   54,   16,    2;
  1,   49,  182,  210,  154,   70,   18,    2;
  1,   64,  280,  392,  364,  224,   88,   20,    2;
  1,   81,  408,  672,  756,  588,  312,  108,   22,    2;
  1,  100,  570, 1080, 1428, 1344,  900,  420,  130,   24,    2;

Maple

A183157 := proc(n, k) if k =0 then 1; elif k = 1 then n^2 ; else 2*(2*n-k+1)*binomial(n, k)/(k+1) ; end if; end proc: # R. J. Mathar, Jan 06 2011

Mathematica

T[_, 0] = 1; T[n_, 1] := n^2; T[n_, k_] := 2*(2*n - k + 1)*Binomial[n, k] / (k + 1);
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 25 2017 *)

Crossrefs

Cf. A183156 (row sums), A006331 (k=2), A008911 (k=3), A067056 (k=4).

Sequence in context: A101020 A160905 A208612 * A211957 A338397 A063983

Adjacent sequences: A183154 A183155 A183156 * A183158 A183159 A183160

Keyword

nonn,tabl

Author

Abdullahi Umar, Dec 28 2010

Status

approved