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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 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
KEYWORD
easy,nonn,tabf
AUTHOR
Emanuele Munarini, Mar 13 2008
STATUS
approved

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Last modified June 12 17:16 EDT 2024. Contains 373339 sequences. (Running on oeis4.)