A001455 Number of permutations of length n with longest increasing subsequence of length 4.

1, 16, 181, 1821, 17557, 167449, 1604098, 15555398, 153315999, 1538907306, 15743413076, 164161815768, 1744049683213, 18865209953045, 207591285198178, 2321616416280982, 26362085777156567, 303635722412859447, 3544040394934246209, 41881891423602685193

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn

References

• J. M. Hammersley, A few seedings of research, in Proc. Sixth Berkeley Sympos. Math. Stat. and Prob., ed. L. M. le Cam et al., Univ. Calif. Press, 1972, Vol. I, pp. 345-394.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..300
• R. M. Baer and P. Brock, Natural sorting over permutation spaces, Math. Comp. 22 1968 385-410.

Crossrefs

Column k=4 of A047874.

Formula

Recurrence: (n-4)*(n+2)^2*(n+3)^2*(n+4)*(225*n^5 - 180*n^4 - 1713*n^3 + 1354*n^2 + 3326*n - 1604)*a(n) = (n+2)^2*(6750*n^9 - 4500*n^8 - 128025*n^7 + 28068*n^6 + 758512*n^5 - 184396*n^4 - 1719825*n^3 + 606292*n^2 + 573428*n - 274224)*a(n-1) - (n-1)*(61425*n^10 - 39915*n^9 - 1118034*n^8 + 644778*n^7 + 5929529*n^6 - 4355935*n^5 - 10322152*n^4 + 7841792*n^3 + 4333856*n^2 - 3087760*n - 58944)*a(n-2) + 2*(n-2)^2*(n-1)*(92250*n^8 - 88875*n^7 - 1380300*n^6 + 1835846*n^5 + 4241004*n^4 - 9250339*n^3 + 4259094*n^2 + 1427720*n - 1155840)*a(n-3) - 576*(n-3)^2*(n-2)^3*(n-1)*(225*n^5 + 945*n^4 - 183*n^3 - 2615*n^2 + 1300*n + 1408)*a(n-4). - Vaclav Kotesovec, Mar 15 2014

a(n) ~ 3 * 2^(4*n+9) / (Pi^(3/2) * n^(15/2)). - Vaclav Kotesovec, Mar 15 2014

Extensions

More terms from Alois P. Heinz, Jul 01 2012

Name of the sequence clarified by Vaclav Kotesovec, Mar 18 2014