%I #60 Jan 08 2021 21:00:49

%S 1,2,1,3,1,2,2,3,1,3,1,4,2,2,4,1,3,3,1,5,2,4,2,4,1,5,3,3,1,5,2,6,4,2,

%T 4,1,6,3,5

%N Irregular triangle T read by rows: each row represents a finite (increasing) oscillation contained in the infinite (increasing) oscillation O.

%C The oscillations are bounded affine permutations. For the definition of a bounded affine permutation, see Definitions 1 and 2 in Madras and Troyka.

%C The infinite (increasing) oscillation O is described by the function f defined as f(s) = s - 4*(-1)^s - 2 with s in the set of integers, while the finite (increasing) oscillations are indecomposable permutations, i.e., that are not the sum of two permutations of nonzero size, and that are contained in O.

%C For each m >= 3, there are exactly two oscillations of size m: 312 and 231, 3142 and 2413, and so on (see p. 22 of Madras and Troyka).

%H Michael H. Albert, Robert Brignall, Vincent Vatter, <a href="https://arxiv.org/abs/1212.3346">Large infinite antichains of permutations</a>, arXiv:1212.3346 [math.CO], 2012; Pure Mathematics and Applications, 24(2) pp. 47-57 (2013).

%H Neal Madras, Justin M. Troyka, <a href="https://arxiv.org/abs/2003.00267"> Bounded affine permutations I. Pattern avoidance and enumeration</a>, arXiv:2003.00267 [math.CO], 2020.

%H Vincent Vatter, <a href="https://arxiv.org/abs/0712.4006">Small permutation classes</a>, arXiv:0712.4006 [math.CO], 2007; Proc. Lond. Math. Soc. 103 (2011), 879-921.

%F T(n, 1) = A158478(n).

%e 1

%e 2 1

%e 3 1 2

%e 2 3 1

%e 3 1 4 2

%e 2 4 1 3

%e 3 1 5 2 4

%e 2 4 1 5 3

%e 3 1 5 2 6 4

%e 2 4 1 6 3 5

%Y Cf. A000142, A266977, A158478, A333616 (row sums).

%K nonn,tabf,more

%O 1,2

%A _Stefano Spezia_, Mar 24 2020