A333416 Irregular triangle T read by rows: each row represents a finite (increasing) oscillation contained in the infinite (increasing) oscillation O.

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, 4, 1, 6, 3, 5

Offset: 1

Author

Stefano Spezia, Mar 24 2020

Keywords

• nonn
• tabf
• more

Comments

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

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.

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).

Examples

			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  4  1  6  3  5
		

Links

• Michael H. Albert, Robert Brignall, Vincent Vatter, Large infinite antichains of permutations, arXiv:1212.3346 [math.CO], 2012; Pure Mathematics and Applications, 24(2) pp. 47-57 (2013).
• Neal Madras, Justin M. Troyka, Bounded affine permutations I. Pattern avoidance and enumeration, arXiv:2003.00267 [math.CO], 2020.
• Vincent Vatter, Small permutation classes, arXiv:0712.4006 [math.CO], 2007; Proc. Lond. Math. Soc. 103 (2011), 879-921.

Crossrefs

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

Formula

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