A201540 -id:A201540 - cp's OEIS frontend

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

A141243 Number of ways to place non-attacking knights on the n X n board.

Original entry on oeis.org

1, 2, 16, 94, 1365, 55213, 3368146, 394631712, 101693175442, 50929053498909, 48988729226134301, 96325314726538906164, 375615195988659173454092, 2933480442104347575000834468, 45480806737377995771543610802659, 1422902021111889804120495149240353936

Offset: 0

Max Alekseyev, Jun 17 2008

The maximum number of non-attacking knights is given by A030978.

Also the number of vertex covers and independent vertex sets in the n X n knight graph.

Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 69-72, 283-285.
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Knight Graph
Eric Weisstein's World of Mathematics, Vertex Cover

Cf. A182407, A182408, A201540.

Row sums of A244081.

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]]];
a[n_] := Function[p, Sum[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, Array[True &, n*3]]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 10}] (* Jean-François Alcover, Mar 29 2016, after Alois P. Heinz's code for A244081 *)

A141243(n=4, s=n^2-1, bad=0)={ while(s && bittest(bad, s), s--);
   if(s < n, 2^(s+1-hammingweight(bad % (2<A141243(n, s-1, bad), x = s%n);
      x > 1 && bad = bitor(bad, 2^(s-n-2)); x < n-2 && bad = bitor(bad, 2^(s-n+2));
      if( s >= 2*n, x && bad = bitor(bad, 2^(s-2*n-1));
              x < n-1 && bad = bitor(bad, 2^(s-2*n+1))
   ); cnt + A141243(n, s-1, bad))} \\ M. F. Hasler, Mar 18 2025

def A141243(n=4, start=(1,1), forbidden=()):
    if start[0] >= n: return 2**sum((n,y+1) not in forbidden for y in range(n))
    nxt = (start[0],start[1]+1) if start[1]A141243(n, nxt, forbidden)
    if start in forbidden: return cnt
    forbidden = {s for s in forbidden if s >= nxt}
    if start[1]2: forbidden |= {(start[0]+1,start[1]-2)}
    if start[0]1: forbidden |= {(start[0]+2,start[1]-1)}
    return cnt+A141243(n, nxt, forbidden) # M. F. Hasler, Mar 17 2025

a(8)-a(13) from R. H. Hardin, Aug 25 2008
a(0) from Alois P. Heinz, Jun 19 2014
a(14) from Hiroaki Yamanouchi, Aug 28 2014
a(15) from Hiroaki Yamanouchi, Aug 29 2014

A244081 Number T(n,k) of ways to place k nonattacking knights on an n X n board; triangle T(n,k), n>=0, 0<=k<=A030978(n), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 6, 4, 1, 1, 9, 28, 36, 18, 2, 1, 16, 96, 276, 412, 340, 170, 48, 6, 1, 25, 252, 1360, 4436, 9386, 13384, 12996, 8526, 3679, 994, 158, 15, 1, 1, 36, 550, 4752, 26133, 97580, 257318, 491140, 688946, 716804, 556274, 323476, 141969, 47684, 12488, 2560, 393, 40, 2

Offset: 0

Alois P. Heinz, Jun 19 2014

In other words, the n-th row gives the coefficients of the independence polynomial of the n X n knight graph. - Eric W. Weisstein, May 05 2017

			T(4,8) = 6:
  ._______. ._______. ._______. ._______. ._______. ._______.
  |_|o|_|o| |o|_|o|_| |o|o|o|o| |o|_|_|o| |o|_|o|o| |o|o|_|o|
  |o|_|o|_| |_|o|_|o| |_|_|_|_| |o|_|_|o| |_|_|_|o| |o|_|_|_|
  |_|o|_|o| |o|_|o|_| |_|_|_|_| |o|_|_|o| |o|_|_|_| |_|_|_|o|
  |o|_|o|_| |_|o|_|o| |o|o|o|o| |o|_|_|o| |o|o|_|o| |o|_|o|o| .
.
Triangle T(n,k) begins:
  1;
  1,  1;
  1,  4,   6,    4,    1;
  1,  9,  28,   36,   18,    2;
  1, 16,  96,  276,  412,  340,   170,    48,    6;
  1, 25, 252, 1360, 4436, 9386, 13384, 12996, 8526, 3679, 994, 158, 15, 1;
  ...
As independence polynomials:
  1
  1 + x
  1 + 4*x + 6*x^2 + 4*x^3 + x^4
  1 + 9*x + 28*x^2 + 36*x^3 + 18*x^4 + 2*x^5
  1 + 16*x + 96*x^2 + 276*x^3 + 412*x^4 + 340*x^5 + 170*x^6 + 48*x^7 + 6*x^8
  ...

Alois P. Heinz, Rows n = 0..12, flattened
Eric Weisstein's World of Mathematics, Independence Polynomial
Eric Weisstein's World of Mathematics, Knight Graph
Eric Weisstein's World of Mathematics, Knights Problem

Columns k=0-6 give: A000012, A000290, A172132, A172134, A172135, A172136, A178499.
T(n,n) gives A201540.
Row sums give A141243.
Cf. A030978.

b:= proc(n, l) option remember; local d, f, g, k;
      d:= nops(l)/3; f:=false;
      if n=0 then 1
    elif l[1..d]=[f$d] then b(n-1, [l[d+1..3*d][], true$d])
    else for k while not l[k] do od; g:= subsop(k=f, l);
         if k>1 then g:=subsop(2*d-1+k=f, g) fi;
         if k2 then g:=subsop(  d-2+k=f, g) fi;
         if k(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n, [true$(n*3)])):
seq(T(n), n=0..7);

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]]];
Table[T[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Mar 28 2016, after Alois P. Heinz *)
Table[Count[IndependentVertexSetQ[KnightTourGraph[n, n], #] & /@ Subsets[Range[n^2], {k}], True], {n, 4}, {k, 0, If[n == 2, 4, (1 - (-1)^n + 2 n^2)/4]}] // Flatten (* Eric W. Weisstein, May 05 2017 *)

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

A243281 Number of inequivalent (mod D_8) ways to place n nonattacking knights on an n X n board.

Original entry on oeis.org

1, 2, 9, 66, 1244, 32524, 1114549, 47513547, 2401002993, 140026612346

Offset: 1

Heinrich Ludwig, Jun 19 2014

Table of n, a(n) for n = 1..10

Cf. A201540, A243716.

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.

Showing 1-8 of 8 results.

cp's OEIS Frontend

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

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A141243 Number of ways to place non-attacking knights on the n X n board.

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A244081 Number T(n,k) of ways to place k nonattacking knights on an n X n board; triangle T(n,k), n>=0, 0<=k<=A030978(n), read by rows.

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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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A243281 Number of inequivalent (mod D_8) ways to place n nonattacking knights on an n X n board.

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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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