

A121632


Triangle read by rows: T(n,k) is the number of deco polyominoes of height n such that the bottom of the last column is at level k (n>=1; k>=0). A deco polyomino is a directed columnconvex polyomino in which the height, measured along the diagonal, is attained only in the last column.


2



1, 2, 5, 1, 16, 7, 1, 65, 43, 11, 1, 326, 279, 98, 16, 1, 1957, 1999, 867, 194, 22, 1, 13700, 15949, 8068, 2225, 348, 29, 1, 109601, 141291, 80493, 25868, 5009, 580, 37, 1, 986410, 1381219, 865728, 313305, 70949, 10229, 913, 46, 1, 9864101, 14798599
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Row n has n1 terms (n>=2). Row sums are the factorials (A000142). T(n,0)=A000522(n1). Sum(k*T(n,k), k>=0)=A121633(n).
T(n,k)=number of permutations of {1,2,...,n} that have k lefttoright maxima not in the initial string of consecutive lefttoright maxima. Example: T(4,1)=7 because we have (13)24, (3)124, (3)142, (2)143, (23)14, (3)214 and (3)241; in each of these permutations 4 is the only lefttoright maximum not in the initial string of lefttoright maxima (shown between parentheses). T(4,2)=1 because we have 2134.  Emeric Deutsch, Apr 04 2008


REFERENCES

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 2942.


LINKS

Table of n, a(n) for n=1..48.


FORMULA

The row generating polynomials satisfy P(n,t)=1t+(t+n1)P(n1,t) for n>=2 with P(1,t)=1. T(n,k)=T(n1,k1)+(n1)T(n1,k) for k>=2.


EXAMPLE

T(2,0)=2 because the deco polyominoes of height 2 are the vertical and horizontal dominoes; the last column of each starts at level 0.
Triangle starts:
1;
2;
5,1;
16,7,1;
65,43,11,1;
326,279,98,16,1;


MAPLE

P[1]:=1: for n from 2 to 12 do P[n]:=sort(expand(1+t*(P[n1]1)+(n1)*P[n1])) od: 1; for n from 2 to 11 do seq(coeff(P[n], t, j), j=0..n2) od; # yields sequence in triangular form


CROSSREFS

Cf. A000142, A000522, A121633.
Sequence in context: A122104 A216121 A104546 * A186361 A197365 A121579
Adjacent sequences: A121629 A121630 A121631 * A121633 A121634 A121635


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Aug 12 2006


STATUS

approved



