A137774 Number of ways to place n nonattacking empresses on an n X n board.
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
Links
- 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
Crossrefs
Formula
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 0
a(n)/n! -> 1/e^4 for 0
Extensions
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
A225554 Longest checkmate in king and empress versus king endgame on an n X n chessboard.
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
Keywords
Comments
An empress moves like a rook and a knight.
Examples
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.
Links
- 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).
Formula
Conjecture: a(n) ~ 7*n/4.
Comments