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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.

%I A137278 #8 Mar 08 2015 18:41:06
%S A137278 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,
%T A137278 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,
%U A137278 72,63,50,37,21,7,1,1,8,28,58,88,118,146,164,171,164,146,118,88,58,28,8,1
%N A137278 Triangle read by rows: g(n,k) = number of ideals of size k in a garland (or double fence) of order n.
%C A137278 Row n has 2n+1 terms.
%C A137278 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
%C A137278 Bounded variation linear paths are path formed from steps 0,1,-1 where the step successions (-1,1) or (1,-1) are not allowed.
%C A137278 Equivalently ternary strings of length n with subwords (0,2) and (2,0) not allowed and total sum k.
%D A137278 T. S. Blyth and J. C. Varlet, Ockham Algebras, Oxford Science Pub. 1994.
%D A137278 E. Munarini, Enumeration of order ideals of a garland, Ars Combin. 76 (2005), 185-192.
%H A137278 Emanuele Munarini, Mar 13 2008, <a href="/A137278/b137278.txt">Table of n, a(n) for n = 0..440</a> [Rows 0 through 20, flattened]
%F A137278 G.f.: G(x,t) = (1-x^2*t^2)/(1-(1+x+x^2)*t+x^2*t^2+x^3*t^3).
%F A137278 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).
%e A137278 In the garland
%e A137278 5..6..7..8
%e A137278 o..o..o..o
%e A137278 |\/|\/|\/|
%e A137278 |/\|/\|/\|
%e A137278 o..o..o..o
%e A137278 1..2..3..4
%e A137278 the ideals of size 4 are 1234, 1253, 1254, 1236, 2347, 1348, 2348.
%e A137278 The ternary strings of size 4 with total sum 4 are
%e A137278 0022, 0202, 0220, 2002, 2020, 2200,
%e A137278 0112, 0121, 0211,
%e A137278 1012, 1021, 2011,
%e A137278 1102, 1201, 2101,
%e A137278 1120, 1210, 2110,
%e A137278 1111
%e A137278 Applying the restriction gives 7 possible strings
%e A137278 0112, 0121, 1012, 2101, 1210, 2110, 1111
%e A137278 Triangle begins:
%e A137278 1,
%e A137278 1, 1, 1,
%e A137278 1, 2, 1, 2, 1,
%e A137278 1, 3, 3, 3, 3, 3, 1,
%e A137278 1, 4, 6, 6, 7, 6, 6, 4, 1,
%e A137278 1, 5, 10, 12, 14, 15, 14, 12, 10, 5, 1,
%e A137278 1, 6, 15, 22, 27, 32, 33, 32, 27, 22, 15, 6, 1,
%e A137278 ...
%Y A137278 Sequence of row sums is A001333 / A078057.
%K A137278 easy,nonn,tabf
%O A137278 0,6
%A A137278 _Emanuele Munarini_, Mar 13 2008