A201861 -id:A201861 - cp's OEIS frontend

A067965 Number of binary arrangements without adjacent 1's on n X n array connected ne-sw and nw-se.

2, 9, 119, 2704, 177073, 21836929, 6985036032, 4576976735769, 7263963336910751, 24830487842030082304, 198126078679714777857441, 3494153303407491549112098721, 141264727800378056245286463971328, 12779122891585386852029424628087941481, 2628141044813862018744988536642011269669959

Offset: 1

R. H. Hardin, Feb 02 2002

			Neighbors for n=4 (dots represent spaces):
. o..o..o..o
...\/ \/ \/
.../\ /\ /\
. o..o..o..o
...\/ \/ \/
.../\ /\ /\
. o..o..o..o
...\/ \/ \/
.../\ /\ /\
. o..o..o..o

Liang Kai, Table of n, a(n) for n = 1..27 (first 19 terms from Vaclav Kotesovec)
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 69, 417.

Main diagonal of A181212.
Cf. circle A000204, line A000045, arrays: e-w ne-sw nw-se A067963, n-s nw-se A067964, e-w n-s nw-se A066864, e-w ne-sw n-s nw-se A063443, n-s A067966, e-w n-s A006506, nw-se A067962, toruses: bare A002416, ne-sw nw-se A067960, ne-sw n-s nw-se A067959, e-w ne-sw n-s nw-se A067958, n-s A067961, e-w n-s A027683, e-w ne-sw n-s A066866.
Cf. A201861, A212271.

Term a(14) from Vaclav Kotesovec, Dec 06 2011
Term a(15) from Vaclav Kotesovec, Jan 03 2012
Term a(16) from Vaclav Kotesovec, May 01 2012
Term a(17)-a(18) from Vaclav Kotesovec, Aug 13 2016

A137774 Number of ways to place n nonattacking empresses on an n X n board.

Original entry on oeis.org

1, 2, 2, 8, 20, 94, 438, 2766, 19480, 163058, 1546726, 16598282, 197708058, 2586423174, 36769177348, 563504645310, 9248221393974, 161670971937362, 2996936692836754, 58689061747521430, 1210222434323163704, 26204614054454840842, 594313769819021397534, 14086979362268860896282

Offset: 1

Vaclav Kotesovec, Jan 27 2011

An empress moves like a rook and a knight.

Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, Separators - a new statistic for permutations, arXiv:1905.12364 [math.CO], 2019.
Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, On the Sparseness of the Downsets of Permutations via Their Number of Separators, Enumerative Combinatorics and Applications (2021) Vol. 1, No. 3, Article #S2R21.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.685 and 636.
W. Schubert, N-Queens page

Cf. A201513, A000170, A002465, A201540.
Cf. A185085, A051223, A244284, A201511, A201861, A137774, A245011.
Cf. A218244, A002464, A110128, A117574, A089222, A002493.

Asymptotics (Vaclav Kotesovec, Jan 26 2011): a(n)/n! -> 1/e^4.
General asymptotic formulas for number of ways to place n nonattacking pieces rook + leaper[r,s] on an n X n board:
a(n)/n! -> 1/e^2 for 0a(n)/n! -> 1/e^4 for 0

Terms a(16)-a(17) from Vaclav Kotesovec, Feb 06 2011
Terms a(18)-a(19) from Wolfram Schubert, Jul 24 2011
Terms a(20)-a(24) (computed by Wolfram Schubert), Vaclav Kotesovec, Aug 25 2012

A201540 Number of ways to place n nonattacking knights on an n X n board.

Original entry on oeis.org

1, 6, 36, 412, 9386, 257318, 8891854, 379978716, 19206532478, 1120204619108, 74113608972922, 5483225594409823, 448414229054798028, 40154319792412218900, 3906519894750904583838

Offset: 1

Vaclav Kotesovec, Dec 02 2011

a(n) = A244081(n,n). - Alois P. Heinz, Jun 19 2014

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 293.

Cf. A172132, A172134, A172135, A172136, A178499, A244081.

Cf. A244284, A201511, A201861, A201513, A141243.

b[n_, l_] := b[n, l] = Module[{d, f, g, k}, d = Length[l]/3; f = False; Which[n == 0, 1, l[[1 ;; d]] == Array[f&, d], b[n - 1, Join[l[[d + 1 ;; 3*d]], Array[True&, d]]], True, For[k = 1, ! l[[k]], k++]; g = ReplacePart[l, k -> f];
     If[k > 1, g = ReplacePart[g, 2*d - 1 + k -> f]];
     If[k < d, g = ReplacePart[g, 2*d + 1 + k -> f]];
     If[k > 2, g = ReplacePart[g, d - 2 + k -> f]];
     If[k < d - 1, g = ReplacePart[g, d + 2 + k -> f]];
     Expand[b[n, ReplacePart[l, k -> f]] + b[n, g]*x]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, Array[True&, n*3]]];
a[n_] := T[n][[n + 1]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 10}] (* Jean-François Alcover, Mar 29 2016, after Alois P. Heinz's code for A244081 *)

a(n) ~ n^(2n)/n!*exp(-9/2). - Vaclav Kotesovec, Nov 29 2011

a(11) from Alois P. Heinz, Jun 19 2014
a(12)-a(13) from Vaclav Kotesovec, Jun 21 2014
a(14) from Vaclav Kotesovec, Aug 26 2016
a(15) from Vaclav Kotesovec, May 26 2021

A278687 Number of non-equivalent ways to place n non-attacking ferses on an n X n board.

Original entry on oeis.org

1, 1, 7, 92, 2102, 66201, 2651890, 126928146, 7060794663, 447024321962

Offset: 1

Heinrich Ludwig, Dec 02 2016

A fers is a leaper [1, 1].

Rotations and reflections of placements are not counted. If they are to be counted, see A201861.

Wikipedia, Fairy chess piece

Cf. A201861, A278688.

A244284 Number of ways to place n nonattacking zebras on an n X n chessboard.

Original entry on oeis.org

1, 6, 84, 1168, 20502, 525796, 18939708, 802444170, 38934305898, 2170312156170

Offset: 1

Vaclav Kotesovec, Jun 25 2014

Zebra is a (fairy chess) leaper [2,3].

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.363
Wikipedia, Fairy chess piece

Cf. A172137, A172138, A172139, A172140.

Cf. A201540, A201511, A201861, A201513.

a(n) ~ n^(2*n)/n! * exp(-9/2).

A182563 Number of ways to place n non-attacking semi-knights on an n x n chessboard.

Original entry on oeis.org

1, 6, 70, 1289, 33864, 1148760, 47700972, 2344465830, 133055587660, 8559364525414, 615266768106190, 48861588247978827, 4247584874013608724, 401107335066453376830, 40880928693752664368224, 4472281486633326131737868

Offset: 1

Vaclav Kotesovec, May 05 2012

Semi-knight is a semi-leaper [1,2]. Moves of a semi-knight are allowed only in [2,1] and [-2,-1]. See also semi-bishops (A187235).

V. Kotesovec, Non-attacking chess pieces, 6ed, p.289

Cf. A182562, A201540, A201513, A002465, A201511, A197989, A201861.

Asymptotic: a(n) ~ n^(2n)/n!*exp(-3/2).

a(16) from Vaclav Kotesovec, May 24 2021

A244288 Number of binary arrangements of total n 1's, without adjacent 1's on n X n array connected nw-se.

Original entry on oeis.org

1, 1, 5, 57, 1084, 29003, 999717, 42125233, 2096106904, 120194547233, 7799803041491, 564856080384900, 45146219773912540, 3946445378386791157, 374482268128153003615, 38330653031858936914329, 4209191997519328986666624, 493575737047609363968826907

Offset: 0

Vaclav Kotesovec, Jun 25 2014

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.422

Cf. A197989, A201540, A201511, A201861, A201513, A244284.

P(m,n) = sum(k=0, (m+1)\2, binomial(m-k+1,k)*x^k, O(x*x^n))
a(n) = polcoef(P(n,n)*prod(m=1, n-1, P(m,n))^2, n) \\ Andrew Howroyd, Mar 27 2023

a(n) ~ n^(2*n)/n! * exp(-3/2).

a(16) from Vaclav Kotesovec, Sep 04 2016
a(17) from Vaclav Kotesovec, Jun 15 2021
a(0)=1 prepended by Andrew Howroyd, Mar 27 2023

A245011 Number of ways to place n nonattacking princesses on an n X n board.

Original entry on oeis.org

1, 4, 6, 86, 854, 9556, 146168, 2660326, 56083228, 1349544632, 36786865968, 1117327217782

Offset: 1

Vaclav Kotesovec, Sep 16 2014

A princess moves like a bishop and a knight.

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 744.
Wikipedia, Fairy chess piece

Cf. A201513, A000170, A002465, A201540.
Cf. A185085, A051223, A244284, A201511, A201861, A137774.
Cf. A225555.

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cp's OEIS Frontend

A067965 Number of binary arrangements without adjacent 1's on n X n array connected ne-sw and nw-se.

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A137774 Number of ways to place n nonattacking empresses on an n X n board.

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A201540 Number of ways to place n nonattacking knights on an n X n board.

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A278687 Number of non-equivalent ways to place n non-attacking ferses on an n X n board.

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A244284 Number of ways to place n nonattacking zebras on an n X n chessboard.

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A182563 Number of ways to place n non-attacking semi-knights on an n x n chessboard.

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A244288 Number of binary arrangements of total n 1's, without adjacent 1's on n X n array connected nw-se.

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A245011 Number of ways to place n nonattacking princesses on an n X n board.

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