A122104 - OEIS

%I #5 Apr 09 2013 10:59:01

%S 1,2,5,1,16,5,3,65,23,20,10,2,326,119,115,84,57,11,8,1957,719,714,582,

%T 526,310,137,55,34,6,13700,5039,5033,4222,4173,3291,2506,972,748,348,

%U 220,38,30,109601,40319,40312,34026,34454,29792,28055,18723,10613,6745

%N Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and such that the sum of the bottom levels of all columns is k (n>=1, k>=0; informally, the number of the "missing" cells in the right bottom corner of the polyomino). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

%C Row n has 1+floor((n-1)^2/4) terms. Row sums are the factorials (A000142). T(n,0)=A000522(n-1). T(n,1)=(n-1)!-1=A033312(n-1). T(n,2)=(n-1)!-n+1=A005096(n-1) for n>=2. Sum(k*T(n,k), k>=0)=A122105(n).

%D E. Barcucci, S. Brunetti and F. Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.

%D E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

%F The row generating polynomials P[n](t) are given by P[n](t)=Q[n](t,1), where Q[1](t,x)=x and Q[n](t,x) = (1/t)Q[n-1](t,tx)+(n-1)xQ[n-1](t,x) for n>=2.

%e Triangle starts:

%e 1;

%e 2;

%e 5,1;

%e 16,5,3;

%e 65,23,20,10,2;

%e 326,119,115,84,57,11,8;

%p Q[1]:=x: for n from 2 to 10 do Q[n]:=simplify(subs(x=t*x,Q[n-1])/t+(n-1)*x*Q[n-1]) od: for n from 1 to 10 do P[n]:=sort(subs(x=1,Q[n])) od: for n from 1 to 10 do seq(coeff(P[n],t,j),j=0..floor((n-1)^2/4)) od; # yields sequence in triangular form

%Y Cf. A000142, A000522, A033312, A005096, A122105.

%K nonn,tabf

%O 1,2

%A _Emeric Deutsch_, Aug 24 2006

%E Keyword tabf added by _Michel Marcus_, Apr 09 2013