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%I #6 Jul 03 2023 18:19:56
%S 1,1,1,2,3,1,7,10,6,1,29,47,33,10,1,147,265,210,82,15,1,889,1740,1521,
%T 697,171,21,1,6252,13087,12373,6377,1885,317,28,1,50163,111066,112016,
%U 63261,21390,4407,540,36,1,452356,1050608,1118991,680541,255245,60903,9247,863,45,1
%N Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 2-cell columns (n>=1; 0<=k<=n-1). 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 sums are the factorials (A000142). T(n,0)=A121638(n). Sum(k*T(n,k), k=0..n-1)=A121639(n)
%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 are P(n,t)=Q(n,t,1,1), where Q(1,t,x,y)=x, Q(2,t,x,y)=x+ty and Q(n,t,x,y)=Q(n-1,t,ty,1/t)+(x+ty+n-3)Q(n-1,t,1,1) for n>=3.
%e T(2,0)=1 and T(2,1)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having, respectively, 0 and 1 2-cell columns.
%e Triangle starts:
%e 1;
%e 1, 1;
%e 2, 3, 1;
%e 7, 10, 6, 1;
%e 29, 47, 33, 10, 1;
%e ...
%p Q[1]:=x: Q[2]:=x+t*y: for n from 3 to 11 do Q[n]:=sort(expand(subs({x=t*y,y=1/t},Q[n-1])+(x+t*y+n-3)*subs({x=1,y=1},Q[n-1]))) od: for n from 1 to 11 do P[n]:=sort(subs({x=1,y=1},Q[n])) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form
%Y Cf. A000142, A121638, A121639, A121554.
%K nonn,tabl
%O 1,4
%A _Emeric Deutsch_, Aug 13 2006
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