

A137278


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


3



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,6


COMMENTS

Row n has 2n+1 terms.
Also triangle of bounded variation linear paths of length n having final height kn (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), 185192.


LINKS



FORMULA

G.f.: G(x,t) = (1x^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



KEYWORD

easy,nonn,tabf


AUTHOR



STATUS

approved



