A189281 -id:A189281 - cp's OEIS frontend

A110128 Number of permutations p of 1,2,...,n satisfying |p(i+2)-p(i)| not equal to 2 for all 0

1, 1, 2, 4, 16, 44, 200, 1288, 9512, 78652, 744360, 7867148, 91310696, 1154292796, 15784573160, 232050062524, 3648471927912, 61080818510972, 1084657970877416, 20361216987032284, 402839381030339816, 8377409956454452732

Offset: 0

Roberto Tauraso, A. Nicolosi and G. Minenkov, Jul 13 2005

When n is even: 1) Number of ways that n persons seated at a rectangular table with n/2 seats along the two opposite sides can be rearranged in such a way that neighbors are no more neighbors after the rearrangement. 2) Number of ways to arrange n kings on an n X n board, with 1 in each row and column, which are non-attacking with respect to the main four quadrants.
a(n) is also number of ways to place n nonattacking pieces rook + alfil on an n X n chessboard (Alfil is a leaper [2,2]) [From Vaclav Kotesovec, Jun 16 2010]
Note that the conjectured recurrence was based on the 600-term b-file, not the other way round. - N. J. A. Sloane, Dec 07 2022

Rintaro Matsuo, Table of n, a(n) for n = 0..600 (terms up to a(35) from Vaclav Kotesovec)
Manuel Kauers, Guessed recurrence operator of order 24 and degree 64
Vaclav Kotesovec, Mathematica program for this sequence
George Spahn and Doron Zeilberger, Counting Permutations Where The Difference Between Entries Located r Places Apart Can never be s (For any given positive integers r and s), arXiv:2211.02550 [math.CO], 2022.
Roberto Tauraso, The Dinner Table Problem: The Rectangular Case, INTEGERS, vol. 6 (2006), paper A11. arXiv:math/0507293.

Cf. A089222, A002464, A117574, A189281.

Column k=2 of A333706.

A formula is given in the Tauraso reference.
Asymptotic (R. Tauraso 2006, quadratic term V. Kotesovec 2011): a(n)/n! ~ (1 + 4/n + 8/n^2)/e^2.
a(n) ~ exp(-2) * n! * (1 + 4/n + 8/n^2 + 68/(3*n^3) + 242/(3*n^4) + 1692/(5*n^5) + 72802/(45*n^6) + 2725708/(315*n^7) + 16083826/(315*n^8) + 186091480/(567*n^9) + 32213578294/(14175*n^10) + ...), based on the recurrence by Manuel Kauers. - Vaclav Kotesovec, Dec 05 2022

Edited by N. J. A. Sloane at the suggestion of Vladeta Jovovic, Jan 01 2008

Terms a(33)-a(35) from Vaclav Kotesovec, Apr 20 2012

A189282 Number of permutations p of 1,2,...,n satisfying p(i+3)-p(i)<>3 for all 1<=i<=n-3.

Original entry on oeis.org

1, 1, 2, 6, 22, 98, 534, 3414, 25498, 217338, 2080990, 22076030, 256888218, 3252308706, 44497313158, 654139144158, 10281397705242, 172033123244330, 3052895403376110, 57266799403366334, 1132124282036449570, 23524895818926592242

Offset: 0

Vaclav Kotesovec, Apr 19 2011

a(n) is also number of ways to place n nonattacking pieces rook + semi-leaper[3,3] on an n X n chessboard.

Christoph Koutschan, Table of n, a(n) for n = 0..48 (terms 0..30 from Vaclav Kotesovec, terms 31..36 from Manuel Kauers and Christoph Koutschan, terms 37..48 from George Spahn and Doron Zeilberger)
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 644.
Vaclav Kotesovec, Mathematica program for this sequence
George Spahn and Doron Zeilberger, Counting Permutations Where The Difference Between Entries Located r Places Apart Can never be s (For any given positive integers r and s), arXiv:2211.02550 [math.CO], 2022.

Cf. A000255, A189281, A117574.

Asymptotics: a(n)/n! ~ (1 + 5/n + 6/n^2)/e.

A189283 Number of permutations p of 1,2,...,n satisfying p(i+4)-p(i)<>4 for all 1<=i<=n-4.

Original entry on oeis.org

1, 1, 2, 6, 24, 114, 628, 4062, 30360, 255186, 2414292, 25350954, 292378968, 3673917102, 49928069188, 729534877758, 11403682481112, 189862332575658, 3354017704180052, 62654508729565554, 1233924707891272728, 25550498290562247438

Offset: 0

Vaclav Kotesovec, Apr 19 2011

a(n) is also number of ways to place n nonattacking pieces rook + semi-leaper[4,4] on an n X n chessboard.

Vaclav Kotesovec, Table of n, a(n) for n = 0..27 (Updated Jan 19 2019)
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 644.
Vaclav Kotesovec, Mathematica program for this sequence
George Spahn and Doron Zeilberger, Counting Permutations Where The Difference Between Entries Located r Places Apart Can never be s (For any given positive integers r and s), arXiv:2211.02550 [math.CO], 2022.

Cf. A000255, A189281, A189282, A189255.

Asymptotics (V. Kotesovec, Mar 2011): a(n)/n! ~ (1 + 7/n + 12/n^2)/e.

Terms a(26)-a(27) from Vaclav Kotesovec, Apr 20 2012

A189843 Number of ways to place n nonattacking composite pieces rook + semi-rider[2,2] on an n X n chessboard.

Original entry on oeis.org

1, 2, 5, 18, 71, 356, 2097, 14212, 105821, 887576, 8093601, 81310936, 876456695, 10257217440, 127631146697, 1705775408656

Offset: 1

Vaclav Kotesovec, Apr 29 2011

a(n) is also number of permutations p of 1,2,...,n satisfying p(j+2k)-p(j)<>2k for all j>=1, k>=1, j+2k<=n

about semi-pieces see semi-bishop (A187235) and semi-queen (A099152)

V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)
Wikipedia, Fairy chess piece

Cf. A189281, A099152, A189838, A187235

A189284 Number of permutations p of 1,2,...,n satisfying p(i+5)-p(i)<>5 for all 1<=i<=n-5.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 696, 4572, 34260, 290328, 2751480, 28686024, 328764732, 4106158164, 55495145304, 806797105320, 12554890849992, 208164423163908, 3663256621120548, 68188490015132040, 1338490745511631080, 27630826605742438968

Offset: 0

Vaclav Kotesovec, Apr 19 2011

a(n) is also number of ways to place n nonattacking pieces rook + semi-leaper[5,5] on an n X n chessboard.

Vaclav Kotesovec, Table of n, a(n) for n = 0..26 (Updated Jan 19 2019)
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 644.
Vaclav Kotesovec, Mathematica program for this sequence
George Spahn and Doron Zeilberger, Counting Permutations Where The Difference Between Entries Located r Places Apart Can never be s (For any given positive integers r and s), arXiv:2211.02550 [math.CO], 2022.

Cf. A000255, A189281, A189282, A189283, A189256.

Asymptotics (V. Kotesovec, Mar 2011): a(n)/n! ~ (1 + 9/n + 20/n^2)/e.

Terms a(25)-a(26) from Vaclav Kotesovec, Apr 20 2012

A189285 Number of permutations p of 1,2,...,n satisfying p(i+6)-p(i)<>6 for all 1<=i<=n-6.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 4920, 37488, 319644, 3033264, 31784280, 364902480, 4538652840, 61102571376, 885045657564, 13722397569072, 226742901078120, 3977354871110160, 73816786920489720, 1444940702597713008, 29750236302549282948

Offset: 0

Vaclav Kotesovec, Apr 19 2011

a(n) is also number of ways to place n nonattacking pieces rook + semi-leaper[6,6] on an n X n chessboard.

Vaclav Kotesovec, Table of n, a(n) for n = 0..24 (Updated Jan 19 2019)
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 644.
Vaclav Kotesovec, Mathematica program for this sequence
George Spahn and Doron Zeilberger, Counting Permutations Where The Difference Between Entries Located r Places Apart Can never be s (For any given positive integers r and s), arXiv:2211.02550 [math.CO], 2022.

Cf. A000255, A189281, A189282, A189283, A189284, A189271.

Asymptotics (V. Kotesovec, Mar 2011): a(n)/n! ~ (1 + 11/n + 30/n^2)/e.

Generally (for this sequence is d=6): 1/e*(1+(2d-1)/n+d*(d-1)/n^2).

Terms a(23)-a(24) from Vaclav Kotesovec, Apr 21 2012

cp's OEIS Frontend

A110128 Number of permutations p of 1,2,...,n satisfying |p(i+2)-p(i)| not equal to 2 for all 0

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A189282 Number of permutations p of 1,2,...,n satisfying p(i+3)-p(i)<>3 for all 1<=i<=n-3.

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A189283 Number of permutations p of 1,2,...,n satisfying p(i+4)-p(i)<>4 for all 1<=i<=n-4.

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A189843 Number of ways to place n nonattacking composite pieces rook + semi-rider[2,2] on an n X n chessboard.

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A189284 Number of permutations p of 1,2,...,n satisfying p(i+5)-p(i)<>5 for all 1<=i<=n-5.

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A189285 Number of permutations p of 1,2,...,n satisfying p(i+6)-p(i)<>6 for all 1<=i<=n-6.

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