A141843 Triangular array T(n,k) (n >= 1, 1 <= k <= n) read by rows: row n gives the lexicographically earliest solution to the n queens problem, or n zeros if no solution exists. The k-th queen is placed in square (k, T(n, k)).

1, 0, 0, 0, 0, 0, 2, 4, 1, 3, 1, 3, 5, 2, 4, 2, 4, 6, 1, 3, 5, 1, 3, 5, 7, 2, 4, 6, 1, 5, 8, 6, 3, 7, 2, 4, 1, 3, 6, 8, 2, 4, 9, 7, 5, 1, 3, 6, 8, 10, 5, 9, 2, 4, 7, 1, 3, 5, 7, 9, 11, 2, 4, 6, 8, 10, 1, 3, 5, 8, 10, 12, 6, 11, 2, 7, 9, 4, 1, 3, 5, 2, 9, 12, 10

Offset: 1

Author

Colin S. Pearson, Jul 10 2008, Aug 16 2008

Keywords

• nonn
• tabl

Comments

History: In December 2017, Matteo Fischetti and Domenico Salvagnin, using Integer Linear Programming (ILP), found solutions for n=56 to n=61; they also found solutions for higher n, but not in contiguous sequence.

Solutions for n=48 to n=55 were found by Wolfram Schubert, around 2010, but not entered in the OEIS.

Entries for board size 46 X 46 (a new solution) and for board size 47 X 47 (already known to Colin Pearson) were added to this sequence in November 2011.

The solution for the 46 X 46 board was discovered by Matthias Engelhardt on Apr 30 2011, the solution for the 47 X 47 board by Colin S. Pearson on Jan 09 2008.

Is it known that the rows converge to (1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 19, 7, 22, 10, 25, 27, 29, 31, ...) ? - M. F. Hasler, Jan 20 2019.

Comments from Don Knuth, Jul 23 2019: (Start)

(i) Concerning the above question about convergence, note that (a) this is sequence A065188, and (b) such convergence is obvious.

(ii) The new 2019 paper by Fischetti and Salvagnin includes solutions for n = 56-61, 63, 65, 67. 69. 71, 73, 77, 79, 85. 91. 93, 97, 101, 103, 109, 115; so the first unknown case is currently n=62.

(iii) I think this sequence (A141843) ought to be the "archival repository" for progress on this problem (I mean, the lexicographically first solutions to the n queens problem). However, it does not yet include Schubert's previously known solutions for n between 48 and 55. Somebody should now add the Fischetti/Salvagnin results too. Each of these is a significant benchmark example, because it's not easy to prove that a partial n-queens solution cannot be extended.

(iv) Here's an excerpt from a message that Matthias Engelhardt sent me on 30 Oct 2017:

> I do not know of a real paper where it is published; up to know, I

> thought it is contained in the OEIS (Online encyclopedia of integer

> sequences, URL http://oeis.org/A141843), but I detect now that the last

> updates are not done! It was Wolfram Schubert who did

> the last computations, and I thought he would update the OEIS. The

> current entry contains the solutions only to n=47. The result for n=48,

> 54 and 55 were computed by him only, the results for n=46, 50, 51, 52,

> 53 were computed by him and verified with my completely different program.

>

> I know I got the results from Wolfram; unfortunately, I cannot find them

> directly. Implicitly, they are contained in the big GIF image which I

> constructed and which is in the internet under

> http://www.nqueens.de/images/firstAlfa.gif. It is linked on my page

> http://www.nqueens.de/sub/FirstAlfa.en.html (I detected also that I must

> correct a header line there).

>

> I think I should update the OEIS in the next days.

(v) Schubert's result for n=48 hasn't been verified independently, as far as I know. It was too hard for Fischetti and Salvagnin's integer-programming approach, and it's also too hard for my Algorithm X. Maybe a SAT solver would verify it though... .(End)

Examples

			Triangle begins:
n\k  [1]  [2]  [3]  [4]  [5]  [6]  [7]  [8]  [9]  [10] [11] [12]
[1]  1;
[2]  0,   0;
[3]  0,   0,   0;
[4]  2,   4,   1,   3;
[5]  1,   3,   5,   2,   4;
[6]  2,   4,   6,   1,   3,   5;
[7]  1,   3,   5,   7,   2,   4,   6;
[8]  1,   5,   8,   6,   3,   7,   2,   4;
[9]  1,   3,   6,   8,   2,   4,   9,   7,   5;
[10] 1,   3,   6,   8,   10,  5,   9,   2,   4,   7;
[11] 1,   3,   5,   7,   9,   11,  2,   4,   6,   8,   10;
[12] 1,   3,   5,   8,   10,  12,  6,   11,  2,   7,   9,   4;
[13] ...
For n=8, the lexicographically smallest solution for the 8-queens problem is 1,5,8,6,3,7,2,4.
		

Links

• Matthias Engelhardt, Table of n, a(n) for n = 1..1892 (previous version of Colin Pearson enlarged)
• Matthias R. Engelhardt, The old nQueens problem.
• Matteo Fischetti, Domenico Salvagnin, Chasing First Queens by Integer Programming, 2018.
• Matteo Fischetti, Domenico Salvagnin, Finding First and Most-Beautiful Queens by Integer Programming, arXiv:1907.08246 [cs.DS], 18 Jul 2019.
• Colin S. Pearson, CSP Queens - Counting Queen-placements.
• Martin S. Pearson, Queens On A Chessboard.
• Wikipedia, Eight Queens Puzzle.

Crossrefs

Cf. A065188, A140450, A000170.

Programs

PARI

row(n)={my(ok(p,a,d)=!for(j=1,n, bittest(d,p[j]-j+n)&& return; bittest(a,p[j]+j)&& return; d+=1<<(p[j]-j+n); a+=1<<(p[j]+j))); for(i=if(n>2,n-3)!*n,n!, ok(numtoperm(n,i))&& return(numtoperm(n,i))); vector(n)} \\ M. F. Hasler, Jan 20 2019

Formula

Limit_{n->oo} Sum_{k=1..n} T(n,k)*x^k = A065188(x).

Extensions

We extended this sequence by adding new terms 1036 to 1128 relating to two further puzzle solutions, for board size 46 X 46 (a new solution) and for board size 47 X 47. Given that the k-th queen is placed in square (k, a(n, k)), we have added the terms (1, a(46, 1)) to (47, a(47, 47)). - Colin S. Pearson, Nov 04 2011

Comments rewritten by Matthias Engelhardt, Jan 28 2018