A078628
Number of ways of arranging the numbers 1..n in a circle so that there is no consecutive triple i, i+1, i+2 or i, i-1, i-2 (mod n).
Original entry on oeis.org
1, 1, 0, 4, 12, 76, 494, 3662, 30574, 284398, 2918924, 32791604, 400400062, 5281683678, 74866857910, 1135063409918, 18330526475060, 314169905117860, 5695984717957246, 108921059813769710, 2190998123920252622, 46250325111346491694
Offset: 1
a(4) = 4: 4 2 1 3, 4 3 1 2, 4 1 3 2, 4 2 3 1.
a(5) = 12: 5 3 1 2 4, 5 2 3 1 4, 5 4 2 1 3, 5 2 4 1 3, 5 1 4 2 3, 5 2 1 4 3, 5 1 3 4 2, 5 3 1 4 2, 5 4 1 3 2, 5 3 4 1 2, 5 2 4 3 1, 5 3 2 4 1.
- Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, http://math.ku.edu/~ilambert/CN.pdf, 2012. - From N. J. A. Sloane, Sep 14 2012
- Isaac Lambert, Table of n, a(n) for n = 1..50
- Wayne M. Dymáček and Isaac Lambert, Circular Permutations Avoiding Runs of i, i+1, i+2 or i, i-1, i-2, Journal of Integer Sequences, Vol. 14 (2011), Article 11.1.6.
- Sean A. Irvine, Java program (github)
- N. J. A. Sloane, FORTRAN program
- Index entries for sequences related to shoe lacings
A095816
Number of permutations of 1..n with no three elements in correct or reverse order.
Original entry on oeis.org
1, 1, 2, 4, 18, 92, 570, 4082, 33292, 304490, 3086890, 34357812, 416526730, 5463479106, 77094352076, 1164544912938, 18749754351338, 320544941916628, 5799226664694602, 110695180631374114, 2223242026407894732, 46868311165318977130, 1034758905785710599402
Offset: 0
- Andrew Howroyd, Table of n, a(n) for n = 0..200
- W. M. Dymacek and I. Lambert, Permutations Avoiding Runs of i, i+1, i+2 or i, i-1, i-2, J. Int. Seq. 14 (2011) # 11.1.6, Table 1.
- D. M. Jackson and R. C. Read, A note on permutations without runs of given length, Aequationes Math. 17 (1978), no. 2-3, 336-343.
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seq(n)={my(m=3); Vec(sum(k=0, n, k!*((2*x^m-x^(m+1)-x)/(x^m-1) + O(x*x^n))^k))} \\ Andrew Howroyd, Aug 31 2018
More terms from Ivana Jovovic (ivana121(AT)EUnet.yu), Nov 11 2007
A165964
Number of circular permutations of length n without increasing or decreasing 3-sequences.
Original entry on oeis.org
1, 1, 0, 4, 16, 86, 542, 3932, 32330, 297438, 3028320, 33814454, 410954878, 5400878692, 76329470882, 1154445436334, 18606430004984, 318369275913710, 5764046146341198, 110091446931897180, 2212282487296335866, 46658484076867264702, 1030533208360458081232
Offset: 1
For n=4 the a(4)=4 solutions are (0,1,3,2), (0,2,1,3), (0,2,3,1), and (0,3,1,2).
- Alois P. Heinz, Table of n, a(n) for n = 1..450
- W. M. Dymacek, I. Lambert, Permutations Avoiding Runs of i, i+1, i+2 or i, i-1, i-2 , J. Int. Seq. 14 (2011) # 11.1.6, Table 1.
A174077
Number of permutations of length n with no consecutive triples i,i+2,i+4 (mod n) or i,i-2,i-4 (mod n).
Original entry on oeis.org
1, 1, 2, 0, 24, 80, 504, 3794, 31616, 290970, 2973600, 33311520, 405781344, 5342413414, 75612197528, 1144942063230, 18471128518656, 316309310084728, 5730646943736936
Offset: 0
As an example, (0,4,1,2,3) is counted by a(5), but (0,4,1,3,2) is not because it has the progression 4,1,3.
A174086
Number of permutations of length n with no consecutive triples i,...i+r,...i+2r (mod n) for all r, and for all equal spacings d.
Original entry on oeis.org
16, 40, 204, 840, 6272, 35856, 378000, 2638460, 28387728, 249444936, 3275745564, 30770034480
Offset: 4
For n=4 note a(4) is the same as the value in A165963 since there are no other distances that can be used (i.e. only d=1).
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