|
Search: id:A121632
|
|
|
| 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 column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. |
|
+0
3
|
|
| 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; listen)
|
|
|
OFFSET
|
1,2
|
|
|
COMMENT
|
Row n has n-1 terms (n>=2). Row sums are the factorials (A000142). T(n,0)=A000522(n-1). Sum(k*T(n,k), k>=0)=A121633(n).
T(n,k)=number of permutations of {1,2,...,n} that have k left-to-right maxima not in the initial string of consecutive left-to-right 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 left-to-right maximum not in the initial string of left-to-right maxima (shown between parentheses). T(4,2)=1 because we have 2134. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 04 2008
|
|
REFERENCES
|
E. Barcucci, A. Del Lungo, and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
|
|
FORMULA
|
The row generating polynomials satisfy P(n,t)=1-t+(t+n-1)P(n-1,t) for n>=2 with P(1,t)=1. T(n,k)=T(n-1,k-1)+(n-1)T(n-1,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[n-1]-1)+(n-1)*P[n-1])) od: 1; for n from 2 to 11 do seq(coeff(P[n], t, j), j=0..n-2) od; # yields sequence in triangular form
|
|
CROSSREFS
|
Cf. A000142, A000522, A121633.
Adjacent sequences: A121629 A121630 A121631 this_sequence A121633 A121634 A121635
Sequence in context: A111797 A122104 A104546 this_sequence A121579 A106852 A120294
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 12 2006
|
|
|
Search completed in 0.002 seconds
|
