A137774 -id:A137774 - cp's OEIS frontend

A189358 Number of permutations p of 1,2,...,n satisfying |p(i+1)-p(i)|<>3 and |p(j+3)-p(j)|<>1 for all i=1..n-1, j=1..n-3.

1, 1, 2, 6, 8, 24, 126, 524, 3072, 22854, 189646, 1827114, 19889946, 238648524, 3131979014, 44540692612, 681114241416, 11136984461270

Offset: 0

Vaclav Kotesovec, Apr 23 2011

a(n) is also the number of ways to place n nonattacking pieces rook + leaper[1,3] on an n X n chessboard (in fairy chess the leaper[1,3] is called a camel).

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

Cf. A137774.

A189358[n_] := Module[{p, c = 0, i = 1, q},
   p=Permutations[Range[n]]; While[i <= Length[p], q = p[[i]]; i++;
    If[AllTrue[Range[n - 1], Abs[q[[# + 1]] - q[[#]]] != 3 &] &&
       AllTrue[Range[n - 3], Abs[q[[# + 3]] - q[[#]]] != 1 &], c++]]; c];
Table[A189358[n], {n, 0, 9}]  (* Robert Price, Apr 04 2019 *)

Asymptotic: a(n)/n! ~ 1/e^4.

a(17) from Alois P. Heinz, Mar 19 2017

A189563 Number of permutations p of 1,2,...,n satisfying |p(i+1)-p(i)|<>4 and |p(j+4)-p(j)|<>1 for all i=1..n-1, j=1..n-4.

Original entry on oeis.org

1, 1, 2, 6, 24, 48, 182, 868, 5752, 37156, 296944, 2738820, 28894206, 335399468, 4285522402, 59536763892, 892785282788

Offset: 0

Vaclav Kotesovec, Apr 23 2011

a(n) is also the number of ways to place n nonattacking pieces rook + leaper[1,4] on an n X n chessboard (in fairy chess the leaper[1,4] is called a giraffe).

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

Cf. A137774, A189358.

Asymptotic: a(n)/n! ~ 1/e^4.

A189564 Number of permutations p of 1,2,...,n satisfying |p(i+1)-p(i)|<>5 and |p(j+5)-p(j)|<>1 for all i=1..n-1, j=1..n-5.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 336, 1474, 8340, 57756, 475658, 4171070, 41950294, 472535256, 5882635676, 79963449714, 1173614446044

Offset: 0

Vaclav Kotesovec, Apr 23 2011

a(n) is also the number of ways to place n nonattacking pieces rook + leaper[1,5] on an n X n chessboard.

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

Cf. A137774, A189358, A189563.

Asymptotic: a(n)/n! ~ 1/e^4.

A225554 Longest checkmate in king and empress versus king endgame on an n X n chessboard.

Original entry on oeis.org

3, 5, 7, 8, 10, 11, 13, 14, 16, 18, 19, 21, 23, 25, 26, 28, 30, 32, 34, 35, 37, 39, 41, 43, 44, 46, 48, 50, 52, 53, 55, 57, 59, 61, 63, 64, 66, 68

Offset: 3

Vaclav Kotesovec, May 10 2013

nonn

An empress moves like a rook and a knight.

			Longest win on an 8x8 chessboard: Ka1 EMb1 - Kd4, 1.Ka1-a2 Kd4-e5 2.Ka2-b3 Ke5-f4 3.Kb3-c3 Kf4-e5 4.EMb1-b6! Ke5-f4 5.Kc3-d4 Kf4-g5 6.Kd4-e4 Kg5-g4! 7.EMb6-e6 Kg4-g3! 8.EMe6-f4! Kg3-h2! 9.Ke4-f3! Kh2-g1! 10.Kf3-g3 Kg1-h1! 11.EMf4-f1#, therefore a(8) = 11.

V. Kotesovec, King and Two Generalised Knights against King, ICGA Journal, Vol. 24, No. 2, pp. 105-107 (2001)
V. Kotesovec, Fairy chess endings on an n x n chessboard, Electronic edition of chess booklets by Vaclav Kotesovec, vol. 8, p.363 (2013), p. 543 (second edition, 2017).

Cf. A137774, A218244, A225551, A225552, A225553, A225555, A225556, A225557, A227437.

Conjecture: a(n) ~ 7*n/4.

A189565 Number of permutations p of 1,2,...,n satisfying |p(i+2)-p(i)|<>3 and |p(j+3)-p(j)|<>2 for all i=1..n-2, j=1..n-3.

Original entry on oeis.org

1, 1, 2, 6, 12, 36, 174, 708, 4334, 31424, 263732, 2503296, 26844578, 316692056, 4090634212, 57274447458, 863488976620

Offset: 0

Vaclav Kotesovec, Apr 23 2011

a(n) is also the number of ways to place n nonattacking pieces rook + leaper[2,3] on an n X n chessboard (in fairy chess the leaper [2,3] is called a zebra).

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

Cf. A137774, A189358, A189563, A189564.

Asymptotic: a(n)/n! ~ 1/e^4.

A189566 Number of permutations p of 1,2,...,n satisfying |p(i+2)-p(i)|<>4 and |p(j+4)-p(j)|<>2 for all i=1..n-2, j=1..n-4.

Original entry on oeis.org

1, 2, 6, 24, 60, 208, 1184, 7840, 51636, 410272, 3836456, 39971896, 455888312, 5717233896, 78164908748, 1153568477544, 18263732340736, 308795344195456, 5550690255143992, 105653899427070440, 2122307518838927952

Offset: 1

Vaclav Kotesovec, Apr 23 2011

a(n) is also the number of ways to place n nonattacking pieces rook + leaper[2,4] on an n X n chessboard.

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

Cf. A137774, A189358, A189563, A189564, A189565.

Asymptotic: a(n)/n! ~ 1/e^4.

a(17)-a(21) from Max Alekseyev, Jul 28 2024

A189567 Number of permutations p of 1,2,...,n satisfying |p(i+2)-p(i)|<>5 and |p(j+5)-p(j)|<>2 for all i=1..n-2, j=1..n-5.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 392, 1810, 10400, 72228, 589674, 5196870, 52398658, 588036216, 7274466172, 98024173852, 1427556373892

Offset: 0

Vaclav Kotesovec, Apr 23 2011

a(n) is also the number of ways to place n nonattacking pieces rook + leaper[2,5] on an n X n chessboard.

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

Cf. A137774, A189358, A189563, A189564, A189565, A189566.

Asymptotic: a(n)/n! ~ 1/e^4.

A189568 Number of permutations p of 1,2,...,n satisfying |p(i+3)-p(i)|<>4 and |p(j+4)-p(j)|<>3 for all i=1..n-3, j=1..n-4.

Original entry on oeis.org

1, 1, 2, 6, 24, 80, 326, 1566, 9544, 65036, 518498, 4750006, 48830634, 554929274, 6926227324, 93970452970, 1377573324202

Offset: 0

Vaclav Kotesovec, Apr 23 2011

a(n) is also the number of ways to place n nonattacking pieces rook + leaper[3,4] on an n X n chessboard.

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

Cf. A137774, A189358, A189563, A189564, A189565, A189566, A189567.

Asymptotic: a(n)/n! ~ 1/e^4.

A189569 Number of permutations p of 1,2,...,n satisfying |p(i+3)-p(i)|<>5 and |p(j+5)-p(j)|<>3 for all i=1..n-3, j=1..n-5.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 464, 2274, 13236, 91760, 740562, 6541984, 65632694, 732880076, 8995905626, 120367234946

Offset: 0

Vaclav Kotesovec, Apr 23 2011

a(n) is also the number of ways to place n nonattacking pieces rook + leaper[3,5] on an n X n chessboard.

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

Cf. A137774, A189358, A189563, A189564, A189565, A189566, A189567, A189568.

Asymptotic: a(n)/n! ~ 1/e^4.

A218244 Number of inequivalent (rotationally and reflectively distinct) ways to place n nonattacking empresses on n X n board.

Original entry on oeis.org

1, 1, 1, 3, 6, 21, 75, 415, 2621, 21066, 195485, 2083543, 24744474, 323438322, 4596672672, 70440521310

Offset: 1

Witold Dlugosz, Oct 24 2012

An empress moves like a rook and a knight.

			a(4) = 3: different ways to place 4 nonattacking empresses on a 4 X 4 board:
Xooo Xooo oXoo
oXoo oooX Xooo
ooXo ooXo oooX
oooX oXoo ooXo

V. Kotesovec, Non-attacking chess pieces

Cf. A137774, A002562.

cp's OEIS Frontend

A189358 Number of permutations p of 1,2,...,n satisfying |p(i+1)-p(i)|<>3 and |p(j+3)-p(j)|<>1 for all i=1..n-1, j=1..n-3.

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A189563 Number of permutations p of 1,2,...,n satisfying |p(i+1)-p(i)|<>4 and |p(j+4)-p(j)|<>1 for all i=1..n-1, j=1..n-4.

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A189564 Number of permutations p of 1,2,...,n satisfying |p(i+1)-p(i)|<>5 and |p(j+5)-p(j)|<>1 for all i=1..n-1, j=1..n-5.

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A225554 Longest checkmate in king and empress versus king endgame on an n X n chessboard.

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A189565 Number of permutations p of 1,2,...,n satisfying |p(i+2)-p(i)|<>3 and |p(j+3)-p(j)|<>2 for all i=1..n-2, j=1..n-3.

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A189566 Number of permutations p of 1,2,...,n satisfying |p(i+2)-p(i)|<>4 and |p(j+4)-p(j)|<>2 for all i=1..n-2, j=1..n-4.

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A189567 Number of permutations p of 1,2,...,n satisfying |p(i+2)-p(i)|<>5 and |p(j+5)-p(j)|<>2 for all i=1..n-2, j=1..n-5.

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A189568 Number of permutations p of 1,2,...,n satisfying |p(i+3)-p(i)|<>4 and |p(j+4)-p(j)|<>3 for all i=1..n-3, j=1..n-4.

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A189569 Number of permutations p of 1,2,...,n satisfying |p(i+3)-p(i)|<>5 and |p(j+5)-p(j)|<>3 for all i=1..n-3, j=1..n-5.

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A218244 Number of inequivalent (rotationally and reflectively distinct) ways to place n nonattacking empresses on n X n board.

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