A121692 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and vertical height (i.e., number of rows) k (1 <= k <= n). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

1, 1, 1, 1, 4, 1, 1, 10, 12, 1, 1, 22, 57, 39, 1, 1, 46, 216, 293, 163, 1, 1, 94, 741, 1651, 1664, 888, 1, 1, 190, 2412, 8181, 12458, 11143, 5934, 1, 1, 382, 7617, 37739, 81255, 102558, 87066, 46261, 1, 1, 766, 23616, 166573, 489753, 823597, 941572, 773772, 409149, 1

Offset: 1

Author

Emeric Deutsch, Aug 17 2006

Keywords

• nonn
• tabl

Comments

Row sums are the factorials (A000142).

T(n,1) = 1,

T(n,2) = 3*2^(n-2) - 2 = A033484(n-2) for n >= 2.

T(n,3) = A121693(n).

Sum_{k=1..n} k*T(n,k) = A121694(n).

Examples

			T(2,1)=1 and T(2,2)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having, respectively, 1 and 2 rows.
Triangle starts:
  1;
  1,  1;
  1,  4,   1;
  1, 10,  12,   1;
  1, 22,  57,  39,   1;
  1, 46, 216, 293, 163, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Elena Barcucci, Sara Brunetti and Francesco Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.
• Elena Barcucci, Alberto Del Lungo, and Renzo Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

Crossrefs

Cf. A000142, A033484, A121693, A121694.

T(2n,n) gives A374794.

Programs

Maple

T:=proc(n,k) option remember; if k=1 then 1 elif k=n then 1 elif k>n then 0 else k*T(n-1,k)+2*T(n-1,k-1)+add(T(n-1,j),j=1..k-2) fi: end: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
with(linalg): a:=proc(i,j) if i=j then i elif i>j then 1 else 0 fi end: p:=proc(Q) local n,A,b,w,QQ: n:=degree(Q): A:=matrix(n,n,a): b:=j->coeff(Q,t,j): w:=matrix(n,1,b): QQ:=multiply(A,w): sort(expand(add(QQ[k,1]*t^k,k=1..n)+t*Q)): end: P[1]:=t: for n from 2 to 11 do P[n]:=p(P[n-1]) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form

Mathematica

T[n_, k_] := T[n, k] = Which[k == 1, 1, k == n, 1, k > n, 0, True, k*T[n - 1, k] + 2*T[n - 1, k - 1] + Sum[T[n - 1, j], {j, 1, k - 2}]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 20 2024 *)

Formula

Rec. relation: T(n,1) = 1; T(n,n) = 1; T(n,k) = k*T(n-1,k) + 2*T(n-1,k-1) + Sum_{j =1..k-2} T(n-1,j) for k <= n; T(n,k) = 0 for k > n.

Rec. relation for the row generating polynomials P[n](t): P[1] = t, P[n] = tP[n-1] + (t+t^2+...+t^(n-1))#P[n-1] for n >= 2. Here # stands for the "max-multiplication" of polynomials, a distributive operation, following the rule t^a # t^b = t^max(a,b).

The second Maple program is based on these polynomials.