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
A201540 Number of ways to place n nonattacking knights on an n X n board.
1, 6, 36, 412, 9386, 257318, 8891854, 379978716, 19206532478, 1120204619108, 74113608972922, 5483225594409823, 448414229054798028, 40154319792412218900, 3906519894750904583838
Offset: 1
Comments
a(n) = A244081(n,n). - Alois P. Heinz, Jun 19 2014
Links
- V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 293.
Crossrefs
Programs
-
Mathematica
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 *)
Formula
a(n) ~ n^(2n)/n!*exp(-9/2). - Vaclav Kotesovec, Nov 29 2011
Extensions
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
A244288 Number of binary arrangements of total n 1's, without adjacent 1's on n X n array connected nw-se.
1, 1, 5, 57, 1084, 29003, 999717, 42125233, 2096106904, 120194547233, 7799803041491, 564856080384900, 45146219773912540, 3946445378386791157, 374482268128153003615, 38330653031858936914329, 4209191997519328986666624, 493575737047609363968826907
Offset: 0
Keywords
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..200
- Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.422
Programs
-
PARI
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
Formula
a(n) ~ n^(2*n)/n! * exp(-3/2).
Extensions
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.
1, 4, 6, 86, 854, 9556, 146168, 2660326, 56083228, 1349544632, 36786865968, 1117327217782
Offset: 1
Comments
A princess moves like a bishop and a knight.
Links
- V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 744.
- Wikipedia, Fairy chess piece
Comments