A183156 -id:A183156 - cp's OEIS frontend

A183158 T(n,k) is the number of partial isometries of an n-chain of fix k (fix of alpha is the number of fixed points of alpha).

1, 1, 1, 4, 2, 1, 12, 6, 3, 1, 38, 10, 6, 4, 1, 90, 26, 10, 10, 5, 1, 220, 42, 15, 20, 15, 6, 1, 460, 106, 21, 35, 35, 21, 7, 1, 1018, 170, 28, 56, 70, 56, 28, 8, 1, 2022, 426, 36, 84, 126, 126, 84, 36, 9, 1, 4304, 682, 45, 120, 210, 252, 210, 120, 45, 10, 1

Offset: 0

Abdullahi Umar, Dec 28 2010

			T (4,2) = 6 because there are exactly 6 partial isometries (on a 4-chain) of fix 2, namely: (1,2)-->(1,2); (2,3)-->(2,3); (3,4)-->(3,4); (1,3)-->(1,3); (2,4)-->(2,4); (1,4)-->(1,4) - the mappings are coordinate-wise.
...1.
...1....1.
...4....2....1.
..12....6....3....1.
..38...10....6....4....1.
..90...26...10...10....5....1.
.220...42...15...20...15....6....1.
.460..106...21...35...35...21....7....1.
1018..170...28...56...70...56...28....8....1.
2022..426...36...84..126..126...84...36....9....1.
4304..682...45..120..210..252..210..120...45...10....1.

R. Kehinde, A. Umar, On the semigroup of partial isometries of a finite chain, arXiv:1101.0049

Cf. A183156 (row sums).

A183159 := proc(n) nh := floor(n/2) ; if type(n,'even') then 13*4^nh-12*nh^2-18*nh-10; else 25*4^nh-12*nh^2-30*nh-22; end if; %/3 ; end proc:
A061547 := proc(n) 3*2^n/8 +(-2)^n/24 - 2/3; end proc:
A183158 := proc(n,k) if k = 0 then A183159(n) ; elif k = 1 then A061547(n+1) ; else binomial(n,k) ; end if; end proc: # R. J. Mathar, Jan 06 2011

T[n_, 0] := (51*2^n + (-2)^n - 40)/12 - n*(n + 3);
T[n_, 1] := (9*2^n + (-1)^(n+1)*2^n - 8)/12;
T[n_, k_] := Binomial[n, k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 22 2017 *)

T(n,0)= A183159(n). T(n,1)=A061547(n+1). T(n,k)=binomial(n,k), k > 1.

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)|).

Original entry on oeis.org

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

Abdullahi Umar, Dec 28 2010

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

			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;

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

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

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

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 *)

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.

cp's OEIS Frontend

A183158 T(n,k) is the number of partial isometries of an n-chain of fix k (fix of alpha is the number of fixed points of alpha).

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