A078628 -id:A078628 - cp's OEIS frontend

A095816 Number of permutations of 1..n with no three elements in correct or reverse order.

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

Jonas Wallgren, Jun 08 2004

nonn

Counts permutations with the property that no subsequence i(i+1)(i+2) or (i+2)(i+1)i occurs.

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.

Cf. A002464, A095817, A095818.

Cf. A165963, A165964, A078628. [From Isaac Lambert, Oct 07 2009]

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

G.f. Sum_{n>=0} n!*((2*x^m-x^(m+1)-x)/(x^m-1))^n where m = 3. - Ivana Jovovic ( ivana121(AT)EUnet.yu ), Nov 11 2007
From Vaclav Kotesovec, May 26 2023: (Start)
a(n) ~ n! * (1 - 2/n + 6/n^2 - 28/(3*n^3) - 10/(3*n^4) + 496/(15*n^5) + 1384/(45*n^6) - 79724/(315*n^7) - 259306/(315*n^8) + 3718094/(2835*n^9) + 33233992/(2025*n^10) + ...).
a(n) = (n-3)*a(n-1) + 3*(n-1)*a(n-2) + (2*n-5)*a(n-3) - (n-3)*a(n-4) - (2*n-13)*a(n-5) - (n-8)*a(n-6) + (n-6)*a(n-7).
(End)

More terms from Ivana Jovovic (ivana121(AT)EUnet.yu), Nov 11 2007

a(0)=1 prepended by Max Alekseyev, Jun 14 2011

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

Isaac Lambert, Oct 07 2009

nonn

Circular permutations are permutations whose indices are from the ring of integers modulo n. Increasing 3-sequences are of the form i,i+1,i+2, while decreasing 3-sequences are of the form i,i-1,i-2.

			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.

Cf. A095816, A165963, A078628.

a(n) = A095816(n-1) - 2 * Sum_{i=1..[(n+1)/3]} (A095816(n-3*i) - A095816(n-1-3*i)). [Corrected by Sean A. Irvine, Jul 07 2025]

Edited and more terms added by Max Alekseyev, Jun 14 2011

A078673 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, 2, 6, 38, 247, 1831, 15287, 142199, 1459462, 16395802, 200200031, 2640841839, 37433428955, 567531704959, 9165263237530, 157084952558930, 2847992358978623, 54460529906884855, 1095499061960126311, 23125162555673245847, 511150714875199094358

Offset: 1

N. J. A. Sloane, Dec 16 2002

nonn

Arrangement that differ only in the direction in which the cycle is traversed do not count as different.

Index entries for sequences related to shoe lacings

Apart from initial terms, A078628/2.

More terms from Sean A. Irvine, Jul 10 2025

A235943 Number a(n,k) of positions (cyclic permutations) of circular permutations of [n] with exactly k (unspecified) increasing or decreasing modular runs (3-sequences), with clockwise and counterclockwise traversals counted as distinct; triangle a(n,k) read by rows, 0<=k<=n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 16, 0, 0, 0, 8, 60, 50, 0, 0, 0, 10, 456, 144, 108, 0, 0, 0, 12, 3458, 1078, 294, 196, 0, 0, 0, 14, 29296, 7936, 2240, 512, 320, 0, 0, 0, 16, 275166, 66096, 16200, 4104, 810, 486, 0, 0, 0, 18, 2843980, 611200, 135600, 29200, 6900, 1200, 700, 0, 0, 0, 20

Offset: 0

Paul J. Campbell, Jan 20 2014, with Joe Marasco and Ashish Vikram

Arrangements that differ in the direction in which the cycle is traversed count as different.

Paul J. Campbell, Circular permutations with exactly one modular run (3-sequence), submitted to Journal of Integer Sequences

Paul J. Campbell, Rows n = 0..13, flattened
Paul J. Campbell, Table of rows n = 0..13 of A235943
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.

Cf. A165961, A165964, A165962, A078628, A078673.

Cf. A235937, A235938, A235939, A235940, A235941, A235942.

A235937 Number of circular permutations with exactly one specified increasing or decreasing modular run (3-sequence), with clockwise and counterclockwise traversals not counted as distinct.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 11, 62, 408, 3056, 25821, 242802, 2517303, 28540102, 351383192, 4669815008, 66640974977, 1016522211474, 16507095990251, 284343231536742, 5178919228252440

Offset: 1

Paul J. Campbell, Jan 20 2014, with Joe Marasco and Ashish Vikram

nonn

Arrangements that differ only in the direction in which the cycle is traversed do not count as different.

This sequence is the same as for straight permutations of {0,1,...,n} that begin with {0,1} and end with {n-1,n} but have no increasing or decreasing 3-sequence, viz., the sequence b(0,1...n-2,n-1) in Dymáček and Lambert.

			With specified sequence 123:
a(5) = 1: 12354.
a(6) = 2: 123564, 123645.
a(7) = 11: 1235476, 1235746, 1235764, 1236475, 1236574, 1236745, 1236754, 1237465, 1237546, 1237564, 1237645.

Paul J. Campbell, Circular permutations with exactly one modular run (3-sequence), submitted to Journal of Integer Sequences

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.

Cf. A165961, A165964, A165962, A078628, A078673.

Cf. A235938, A235939, A235940, A235941, A235942, A235943.

a(20)-a(21) from Alois P. Heinz, Jan 24 2014

Obsolete b-file deleted by N. J. A. Sloane, Jan 05 2019

A235938 Number of circular permutations with exactly one specified increasing or decreasing modular run (3-sequence), with clockwise and counterclockwise traversals counted as distinct.

Original entry on oeis.org

0, 0, 0, 0, 2, 4, 22, 124, 816, 6112, 51642, 485604, 5034606, 57080204, 702766384, 9339630016, 133281949954, 2033044422948, 33014191980502, 568686463073484, 10357838456504880

Offset: 1

Paul J. Campbell, Jan 20 2014, with Joe Marasco and Ashish Vikram

nonn

			With specified sequence 123:
a(5) = 2: 12354, 32154.
a(6) = 4: 123564, 321564, 123645, 321546.

Paul J. Campbell, Circular permutations with exactly one modular run (3-sequence), submitted to Journal of Integer Sequences

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.

Cf. A165961, A165964, A165962, A078628, A078673.

Cf. A235937, A235939, A235940, A235941, A235942, A235943.

a(n) = 2*A235937(n).

a(20)-a(21) from Alois P. Heinz, Jan 24 2014

Obsolete b-file deleted by N. J. A. Sloane, Jan 05 2019

A235939 Number of circular permutations with exactly one (unspecified) increasing or decreasing modular 3-sequence, with clockwise and counterclockwise traversals not counted as distinct.

Original entry on oeis.org

0, 0, 0, 0, 5, 12, 77, 496, 3672, 30560, 284031, 2913624, 32724939, 399561428, 5270747880, 74717040128, 1132896574609, 18297399806532, 313634823814769, 5686864630734840, 108757303793301240

Offset: 1

Paul J. Campbell, Jan 20 2014, with Joe Marasco and Ashish Vikram

nonn

Arrangements that differ only in the direction in which the cycle is traversed do not count as different.

			a(5) = 5: 12354, 23415, 34521, 45132, 51243.

Paul J. Campbell, Circular permutations with exactly one modular run (3-sequence), submitted to Journal of Integer Sequences

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.

Cf. A165961, A165964, A165962, A078628, A078673.

Cf. A235937, A235938, A235940, A235941, A235942, A235943.

a(n) = n*A235937(n).

a(20)-a(21) from Alois P. Heinz, Jan 24 2014

Obsolete b-file deleted by N. J. A. Sloane, Jan 05 2019

A235940 Number of circular permutations with exactly one (unspecified) increasing or decreasing modular 3-sequence, with clockwise and counterclockwise traversals counted as distinct.

Original entry on oeis.org

0, 0, 0, 0, 10, 24, 154, 992, 7344, 61120, 568062, 5827248, 65449878, 799122856, 10541495760, 149434080256, 2265793149218, 36594799613064, 627269647629538, 11373729261469680, 217514607586602480

Offset: 1

Paul J. Campbell, Jan 20 2014, with Joe Marasco and Ashish Vikram

Paul J. Campbell, Circular permutations with exactly one modular run (3-sequence), submitted to Journal of Integer Sequences.

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.

Cf. A165961, A165964, A165962, A078628, A078673.

Cf. A235937, A235938, A235939, A235941, A235942, A235943.

a(n) = 2n*A235937(n).
a(n) = n*A235938(n).
a(n) = 2*A235939(n).

a(20)-a(21) added using the data at A235939 by Amiram Eldar, May 06 2024

A235941 Positions (cyclic permutations) of circular permutations with exactly one (unspecified) increasing or decreasing modular 3-sequence, with clockwise and counterclockwise traversals not counted as distinct.

Original entry on oeis.org

0, 0, 0, 0, 25, 72, 539, 3968, 33048, 305600, 3124341, 34963488, 425424207, 5593859992, 79061218200, 1195472642048, 19259241768353, 329353196517576, 5959061652480611, 113737292614696800, 2283903379659326040

Offset: 1

Paul J. Campbell, Jan 20 2014, with Joe Marasco and Ashish Vikram

nonn

Paul J. Campbell, Circular permutations with exactly one modular run (3-sequence), submitted to Journal of Integer Sequences

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.

Cf. A165961, A165964, A165962, A078628, A078673.

Cf. A235937, A235938, A235939, A235940, A235942, A235943.

a(n) = n^2 * A235937(n).

a(n) = n * A235939(n).

a(20)-a(21) from Alois P. Heinz, Jan 24 2014

Obsolete b-file deleted by N. J. A. Sloane, Jan 05 2019

A235942 Number of positions (cyclic permutations) of circular permutations with exactly one (unspecified) increasing or decreasing modular 3-sequence, with clockwise and counterclockwise traversals counted as distinct.

Original entry on oeis.org

0, 0, 0, 0, 50, 144, 1078, 7936, 66096, 611200, 6248682, 69926976, 850848414, 11187719984, 158122436400, 2390945284096, 38518483536706, 658706393035152, 11918123304961222, 227474585229393600, 4567806759318652080

Offset: 1

Paul J. Campbell, Jan 20 2014, with Joe Marasco and Ashish Vikram

nonn

Paul J. Campbell, Circular permutations with exactly one modular run (3-sequence), submitted to Journal of Integer Sequences

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.

Cf. A165961, A165964, A165962, A078628, A078673.

Cf. A235937, A235938, A235939, A235940, A235941, A235943.

a(n) = 2*n^2 * A235937(n).
a(n) = n^2 * A235938(n).
a(n) = 2*n * A235939(n).
a(n) = n * A235940(n).
a(n) = 2 * A235941(n).

a(20)-a(21) from Alois P. Heinz, Jan 24 2014

Obsolete b-file deleted by N. J. A. Sloane, Jan 05 2019

cp's OEIS Frontend

A095816 Number of permutations of 1..n with no three elements in correct or reverse order.

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A165964 Number of circular permutations of length n without increasing or decreasing 3-sequences.

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A078673 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).

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A235943 Number a(n,k) of positions (cyclic permutations) of circular permutations of [n] with exactly k (unspecified) increasing or decreasing modular runs (3-sequences), with clockwise and counterclockwise traversals counted as distinct; triangle a(n,k) read by rows, 0<=k<=n.

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A235937 Number of circular permutations with exactly one specified increasing or decreasing modular run (3-sequence), with clockwise and counterclockwise traversals not counted as distinct.

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A235938 Number of circular permutations with exactly one specified increasing or decreasing modular run (3-sequence), with clockwise and counterclockwise traversals counted as distinct.

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A235939 Number of circular permutations with exactly one (unspecified) increasing or decreasing modular 3-sequence, with clockwise and counterclockwise traversals not counted as distinct.

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A235940 Number of circular permutations with exactly one (unspecified) increasing or decreasing modular 3-sequence, with clockwise and counterclockwise traversals counted as distinct.

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A235941 Positions (cyclic permutations) of circular permutations with exactly one (unspecified) increasing or decreasing modular 3-sequence, with clockwise and counterclockwise traversals not counted as distinct.

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