A189256 -id:A189256 - cp's OEIS frontend

A333706 Number T(n,k) of permutations p of [n] such that |p(i+k) - p(i)| <> k for i in [n-k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 2, 0, 0, 4, 6, 0, 2, 16, 20, 24, 0, 14, 44, 80, 108, 120, 0, 90, 200, 384, 544, 672, 720, 0, 646, 1288, 2240, 3264, 4128, 4800, 5040, 0, 5242, 9512, 15424, 23040, 28992, 34752, 38880, 40320, 0, 47622, 78652, 123456, 176832, 231936, 280512, 323520, 352800, 362880

Offset: 0

Alois P. Heinz, Apr 02 2020

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = n! for k>=n.

			Triangle T(n,k) begins:
  1;
  0,    1;
  0,    0,    2;
  0,    0,    4,     6;
  0,    2,   16,    20,    24;
  0,   14,   44,    80,   108,   120;
  0,   90,  200,   384,   544,   672,   720;
  0,  646, 1288,  2240,  3264,  4128,  4800,  5040;
  0, 5242, 9512, 15424, 23040, 28992, 34752, 38880, 40320;
  ...

Alois P. Heinz, Rows n = 0..20, flattened
Roberto Tauraso, The Dinner Table Problem: The Rectangular Case, INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 6 (2006), #A11.
Wikipedia, Permutation

Columns k=0-10 (for n>=k) give: A000007, A002464, A110128, A117574, A189255, A189256, A189271, A360384, A360386, A360462, A360463.
Main diagonal gives A000142.
T(2n,n) gives A189849.
T(n+1,n) gives 4*A138772(n).
T(n+2,n) gives 16*A333804(n).
Cf. A000170 (condition is satisfied for all k), A248686 (p(i) at distance k are sorted).

A189271 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, 2, 6, 24, 120, 720, 4800, 34752, 280512, 2528256, 25282560, 278323200, 3289036800, 42336448512, 589351062528, 8820501301248, 141215147788800, 2407845089203200, 43543159894318080, 832618225074748416, 16782891792284791296

Offset: 1

Vaclav Kotesovec, Apr 19 2011

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

Vaclav Kotesovec, Table of n, a(n) for n = 1..24
Vaclav Kotesovec, Non-attacking chess pieces, Sixth edition, p. 633, Feb 02 2013.
Vaclav Kotesovec, Mathematica program for this sequence
Roberto Tauraso, The Dinner Table Problem: The Rectangular Case, INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 6 (2006), #A11.

Cf. A002464, A110128, A117574, A189255, A189256.

Column k=6 of A333706.

Asymptotic (R. Tauraso 2006, quadratic term V. Kotesovec 2011): a(n)/n! ~ (1 + 20/n + 168/n^2)/e^2.

Generally (for this sequence is d=6): 1/e^2*(1+4(d-1)/n+2d*(3d-4)/n^2+...).

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

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

A189841 Number of ways to place n nonattacking composite pieces rook + rider[5,5] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 24, 120, 672, 4128, 28992, 231936, 2088960, 20017152, 207208704, 2326900992, 28338241536, 373152276480, 5206300028928

Offset: 1

Vaclav Kotesovec, Apr 29 2011

a(n) is also number of permutations p of 1,2,...,n satisfying |p(j+5k)-p(j)|<>5k for all j>=1, k>=1, j+5k<=n

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

Cf. A189256, A000170, A189838, A189839, A189840

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A333706 Number T(n,k) of permutations p of [n] such that |p(i+k) - p(i)| <> k for i in [n-k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A189271 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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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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A189841 Number of ways to place n nonattacking composite pieces rook + rider[5,5] on an n X n chessboard.

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