A235940 -id:A235940 - cp's OEIS frontend

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.

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

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

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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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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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.

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