A137278 Triangle read by rows: g(n,k) = number of ideals of size k in a garland (or double fence) of order n.

1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 3, 3, 3, 3, 1, 1, 4, 6, 6, 7, 6, 6, 4, 1, 1, 5, 10, 12, 14, 15, 14, 12, 10, 5, 1, 1, 6, 15, 22, 27, 32, 33, 32, 27, 22, 15, 6, 1, 1, 7, 21, 37, 50, 63, 72, 75, 72, 63, 50, 37, 21, 7, 1, 1, 8, 28, 58, 88, 118, 146, 164, 171, 164, 146, 118, 88, 58, 28, 8, 1

Offset

0,6

Comments

Row n has 2n+1 terms.

Also triangle of bounded variation linear paths of length n having final height k-n (height varies from -n to n). Olivier Gérard, Aug 28 2012

Bounded variation linear paths are path formed from steps 0,1,-1 where the step successions (-1,1) or (1,-1) are not allowed.

Equivalently ternary strings of length n with subwords (0,2) and (2,0) not allowed and total sum k.

References

T. S. Blyth and J. C. Varlet, Ockham Algebras, Oxford Science Pub. 1994.

E. Munarini, Enumeration of order ideals of a garland, Ars Combin. 76 (2005), 185-192.

Links

Emanuele Munarini, Mar 13 2008, Table of n, a(n) for n = 0..440 [Rows 0 through 20, flattened]

Formula

G.f.: G(x,t) = (1-x^2*t^2)/(1-(1+x+x^2)*t+x^2*t^2+x^3*t^3).

Recurrence: g(n+3,k+3) = g(n+2,k+3) + g(n+2,k+2) + g(n+2,k+1) - g(n+1,k+1) - g(n,k).

Example

In the garland
5..6..7..8
o..o..o..o
|\/|\/|\/|
|/\|/\|/\|
o..o..o..o
1..2..3..4
the ideals of size 4 are 1234, 1253, 1254, 1236, 2347, 1348, 2348.
The ternary strings of size 4 with total sum 4 are
0022, 0202, 0220, 2002, 2020, 2200,
0112, 0121, 0211,
1012, 1021, 2011,
1102, 1201, 2101,
1120, 1210, 2110,
1111
Applying the restriction gives 7 possible strings
0112, 0121, 1012, 2101, 1210, 2110, 1111
Triangle begins:
1,
1, 1, 1,
1, 2, 1, 2, 1,
1, 3, 3, 3, 3, 3, 1,
1, 4, 6, 6, 7, 6, 6, 4, 1,
1, 5, 10, 12, 14, 15, 14, 12, 10, 5, 1,
1, 6, 15, 22, 27, 32, 33, 32, 27, 22, 15, 6, 1,
...

Crossrefs

Sequence of row sums is A001333 / A078057.

Sequence in context: A059780 A075119 A224076 * A205810 A139368 A266506

Adjacent sequences: A137275 A137276 A137277 * A137279 A137280 A137281

Keyword

easy,nonn,tabf

Author

Emanuele Munarini, Mar 13 2008

Status

approved