A061061 Maximal number of 132 patterns in a permutation of 1,2,...,n.

0, 0, 1, 3, 6, 12, 20, 31, 46, 64, 87, 115, 147, 186, 231, 282, 342, 408, 482, 566, 657, 759, 871, 991, 1126, 1270, 1424, 1594, 1774, 1968, 2177, 2397, 2635, 2887, 3151, 3436, 3735, 4050, 4386, 4736, 5106, 5496, 5901, 6330, 6778, 7244, 7737, 8247, 8778, 9336

Offset

1,4

Comments

a(n) = A216499(n) - (n choose 3). lim_{n --> infinity} a(n) / n^3 = (2 sqrt(3) - 3) / 6 = 0.077350... a(n) / n^3 < (2 sqrt(3) - 3) / 6 = 0.077350... for all n > 0. [Chao et al. (2012)]. - Jesper Jansson, Sep 10 2012

References

K.-M. Chao, A.-C. Chu, J. Jansson, R. S. Lemence, and A. Mancheron. Asymptotic Limits of a New Type of Maximization Recurrence with an Application to Bioinformatics. Proceedings of the Ninth Annual Conference on Theory and Applications of Models of Computation (TAMC 2012), Lecture Notes in Computer Science, Vol. 7287, pp. 177-188, Springer-Verlag Berlin Heidelberg, 2012.

W. Stromquist, Packing layered posets into posets, manuscript.

Links

Table of n, a(n) for n=1..50.

Miklos Bona, Bruce E. Sagan and Vincent R. Vatter, Pattern frequency sequences and internal zeros

Alejandro H. Morales, Igor Pak, Greta Panova, Asymptotics of principal evaluations of Schubert polynomials for layered permutations, arXiv:1805.04341 [math.CO], 2018.

Formula

a(n) = max(a(k) + k*C(n-k, 2): 1 <= k < n)

a(n+1)/a(n)=1+3/n+O(1/n^2). n^2*(a(n+1)/a(n)-1-3/n) is bounded but there is no limit; limit n-->infinity a(n)/n^3 = C = 0.0773... - Benoit Cloitre, Jan 25 2003

Example

a(8) = 31; the permutation of 1..8 containing the maximum number of 132 patterns is 13287654.

Crossrefs

Sequence in context: A246147 A066140 A200067 * A361272 A001975 A096220

Adjacent sequences: A061058 A061059 A061060 * A061062 A061063 A061064

Keyword

easy,nonn

Author

Michael Albert, May 27 2001

Extensions

More terms from Vladeta Jovovic, Jun 03 2001

Status

approved