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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)|).

%I A183157 #23 Feb 16 2025 08:33:13
%S A183157 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,
%T A183157 2,1,49,182,210,154,70,18,2,1,64,280,392,364,224,88,20,2,1,81,408,672,
%U A183157 756,588,312,108,22,2,1,100,570,1080,1428,1344,900,420,130,24,2
%N 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)|).
%C A183157 Rows also give the coefficients of the clique polynomial of the n X n bishop graph. - _Eric W. Weisstein_, Jun 04 2017
%H A183157 R. Kehinde and A. Umar, <a href="http://arxiv.org/abs/1101.2558">On the semigroup of partial isometries of a finite chain</a>, arXiv:1101.2558 [math.GR], 2011.
%H A183157 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BishopGraph.html">Bishop Graph</a>
%H A183157 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CliquePolynomial.html">Clique Polynomial</a>
%F A183157 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.
%e A183157 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.
%e A183157 The triangle starts
%e A183157   1;
%e A183157   1,    1;
%e A183157   1,    4,    2;
%e A183157   1,    9,   10,    2;
%e A183157   1,   16,   28,   12,    2;
%e A183157   1,   25,   60,   40,   14,    2;
%e A183157   1,   36,  110,  100,   54,   16,    2;
%e A183157   1,   49,  182,  210,  154,   70,   18,    2;
%e A183157   1,   64,  280,  392,  364,  224,   88,   20,    2;
%e A183157   1,   81,  408,  672,  756,  588,  312,  108,   22,    2;
%e A183157   1,  100,  570, 1080, 1428, 1344,  900,  420,  130,   24,    2;
%p A183157 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 A183157 T[_, 0] = 1; T[n_, 1] := n^2; T[n_, k_] := 2*(2*n - k + 1)*Binomial[n, k] / (k + 1);
%t A183157 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 25 2017 *)
%Y A183157 Cf. A183156 (row sums), A006331 (k=2), A008911 (k=3), A067056 (k=4).
%K A183157 nonn,tabl
%O A183157 0,5
%A A183157 _Abdullahi Umar_, Dec 28 2010