A172449 -id:A172449 - cp's OEIS frontend

A319284 The profiles of the backtrack tree for the n queens problem, triangle read by rows.

1, 1, 1, 1, 2, 0, 1, 3, 2, 0, 1, 4, 6, 4, 2, 1, 5, 12, 14, 12, 10, 1, 6, 20, 36, 46, 40, 4, 1, 7, 30, 76, 140, 164, 94, 40, 1, 8, 42, 140, 344, 568, 550, 312, 92, 1, 9, 56, 234, 732, 1614, 2292, 2038, 1066, 352, 1, 10, 72, 364, 1400, 3916, 7552, 9632, 7828, 4040, 724, 1, 11, 90, 536, 2468, 8492, 21362, 37248, 44148, 34774, 15116, 2680

Offset: 0

Peter Luschny, Sep 16 2018

The profile (p_0, p_1, ..., p_n) is the number of nodes at each level of the tree.

			[1]
[1,  1]
[1,  2,  0]
[1,  3,  2,    0]
[1,  4,  6,    4,    2]
[1,  5,  12,  14,   12,    10]
[1,  6,  20,  36,   46,    40,     4]
[1,  7,  30,  76,  140,   164,    94,     40]
[1,  8,  42, 140,  344,   568,   550,    312,     92]
[1,  9,  56, 234,  732,  1614,  2292,   2038,   1066,    352]
[1, 10,  72, 364, 1400,  3916,  7552,   9632,   7828,   4040,    724]
[1, 11,  90, 536, 2468,  8492, 21362,  37248,  44148,  34774,  15116,  2680]
[1, 12, 110, 756, 4080, 16852, 52856, 120104, 195270, 222720, 160964, 68264, 14200]

D. E. Knuth, The Art of Computer Programming, Volume 4, Pre-fascicle 5B, Introduction to Backtracking, 7.2.2. Backtrack programming. 2018.

Peter Luschny, Rows n = 0..19, flattened
Candida Bowtell and Peter Keevash, The n-queens problem, arXiv:2109.08083 [math.CO] 2021.
V. Kotesovec, Ways of placing non-attacking queens and kings..., part of "Between chessboard and computer", 1996, pp. 204 - 206.
Peter Luschny, Julia implementation of the n queens problem with profiles
Michael Simkin, The number of n-queens configurations, arXiv:2107.13460 [math.CO] 2021.
Wikipedia, Backtracking
Wikipedia, Eight queens puzzle

Cf. A000170 (T(n,n)), A319283 (row sums), A319288 (indices of the row maxima).

Cf. A000012 (col. 0), A000027 (col. 1), A002378 (col. 2), A061989 and A079908 (col. 3), A061990 (col. 4), A061991 (col. 5), A061992 (col. 6), A061993 (col. 7), A172449 (col. 8).

Julia
```
# See the link section.
```

A174698 Number of ways to place 8 nonattacking knights on an 8 X n board.

Original entry on oeis.org

1, 81, 4409, 175720, 2479881, 17925691, 92952858, 379978716, 1286959255, 3765248749, 9805497200, 23226916560, 50866495373, 104288896551, 202154535834, 373400685738, 661407061211, 1129334088897, 1866838857216

Offset: 1

Vaclav Kotesovec, Mar 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes

Cf. A172449, A172212, A172213, A172214, A172215, A172217.

CoefficientList[Series[(592 x^21 - 584 x^20 - 18100 x^19 + 49628 x^18 + 134264 x^17 - 735838 x^16 + 584418 x^15 + 2607764 x^14 - 7093608 x^13 + 5656936 x^12 + 5136811 x^11 - 13973779 x^10 + 14583702 x^9 - 1612610 x^8 + 2009820 x^7 + 6682287 x^6 + 1572406 x^5 + 1050447 x^4 + 138871 x^3 + 3716 x^2 + 72 x + 1) / (1 - x)^9, {x, 0, 50}], x] (* Vincenzo Librandi, May 30 2013 *)

Explicit formula: a(n) = (262144*n^8 -6881280*n^7+93456384*n^6 -838693632*n^5 +5361604836*n^4 -24739168020*n^3+79766188151*n^2 -163079018193*n +160750559340)/630, n>=14.

G.f.: x*(592*x^21 -584*x^20 -18100*x^19 +49628*x^18+134264*x^17 -735838*x^16 +584418*x^15+2607764*x^14 -7093608*x^13 +5656936*x^12 +5136811*x^11 -13973779*x^10 +14583702*x^9 -1612610*x^8 +2009820*x^7 +6682287*x^6 +1572406*x^5 +1050447*x^4 +138871*x^3+3716*x^2 +72*x+1)/(1-x)^9.

A269133 Number of ways to place m nonattacking queens on an m X n board, 1 <= m <= n (triangular array).

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 4, 6, 4, 2, 5, 12, 14, 12, 10, 6, 20, 36, 46, 40, 4, 7, 30, 76, 140, 164, 94, 40, 8, 42, 140, 344, 568, 550, 312, 92, 9, 56, 234, 732, 1614, 2292, 2038, 1066, 352, 10, 72, 364, 1400, 3916, 7552, 9632, 7828, 4040, 724, 11, 90, 536, 2468, 8492, 21362, 37248, 44148, 34774, 15116, 2680, 12, 110, 756, 4080, 16852, 52856, 120104, 195270, 222720, 160964, 68264, 14200

Offset: 1

Marko Riedel, Feb 19 2016

			The triangular array begins:
   n\m  1   2   3    4     5     6      7      8      9     10    11    12
   1    1
   2    2   0
   3    3   2   0
   4    4   6   4    2
   5    5  12  14   12    10
   6    6  20  36   46    40     4
   7    7  30  76  140   164    94     40
   8    8  42 140  344   568   550    312     92
   9    9  56 234  732  1614  2292   2038   1066    352
  10   10  72 364 1400  3916  7552   9632   7828   4040    724
  11   11  90 536 2468  8492 21362  37248  44148  34774  15116  2680
  12   12 110 756 4080 16852 52856 120104 195270 222720 160964 68264 14200
...

Math StackExchange, State space for eight queen problem
Marko Riedel, Perl program to compute triangular array of nonattacking queens configurations

Cf. A000170 (m=n), A002562, A065256, A348129.
Cf. A000027 (m=1), A002378 (m=2), A061989 (m=3), A061990 (m=4), A061991 (m=5), A061992 (m=6), A061993 (m=7), A172449 (m=8).
Cf. A036464 (2Q), A047659 (3Q), A061994 (4Q), A108792 (5Q), A176186 (6Q).
Cf. A006717, A051906, A319284 (backtrack trees).

{A269133(m, n, B=[], t=if(#B, setminus(n, Set(concat(B+t=[-#B..-1], B-t))), n=[1..n]))= if(#B < m-1, vecsum([A269133(m, setminus(n, [t]), concat(B,t)) | t<-t]), #t)} \\ M. F. Hasler, Jan 11 2022

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A319284 The profiles of the backtrack tree for the n queens problem, triangle read by rows.

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A174698 Number of ways to place 8 nonattacking knights on an 8 X n board.

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A269133 Number of ways to place m nonattacking queens on an m X n board, 1 <= m <= n (triangular array).

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