A319284 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

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

REFERENCES

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

LINKS

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

EXAMPLE

[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]

PROG

(Julia) # See the link section.

CROSSREFS

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).

Sequence in context: A216235 A306914 A317023 * A338526 A182703 A354006

Adjacent sequences: A319281 A319282 A319283 * A319285 A319286 A319287

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Sep 16 2018

STATUS

approved