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

Author

Michael Albert, May 27 2001

Keywords

• easy
• nonn

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

Examples

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

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

• 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

Extensions

More terms from Vladeta Jovovic, Jun 03 2001