This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378488 #6 Dec 12 2024 23:15:21 %S A378488 0,0,0,10,13,10,13,36,44,50,69,75,83,106,109,186,346,373,533,186,346, %T A378488 373,533,980,1032,1090,1108,1188,1244,1399,1515,1519,1905,1956,2074, %U A378488 2090,2197,2210,2390,2649,2829,2842,2949,2965,3083,3134,3520,3524,3640,3795,3851 %N A378488 Table T(n,k) read by rows where in the n-th row the k-th column is the permutation rank of the k-th solution to the n-queens problem in a n X n board. %C A378488 The length of the n-th row is A000170(n) for n >= 4. %H A378488 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eight_queens_puzzle">Eight queens puzzle</a> %F A378488 a(n) = 0 if no solution exists or n = 1. %e A378488 Table T(n,k) reads as follows: %e A378488 n / k %e A378488 ------------------------------------------- %e A378488 1 | 0 %e A378488 2 | 0 %e A378488 3 | 0 %e A378488 4 | 10, 13 %e A378488 5 | 10, 13, 36, 44, 50, 69, 75, 83, 106, 109 %e A378488 6 | 186, 346, 373, 533 %e A378488 For a table of 4 by 4 one of the solutions for placing the 4 queens is [(0,1),(1,3),(2,0),(3,2)] and its compact representation is [1, 3, 0, 2], %e A378488 this resulting representation is a permutation that can be ranked and its rank is 10. %e A378488 T(1) = [0] %e A378488 *-* %e A378488 |Q| Permutation: [0], Rank: 0 %e A378488 *-* %e A378488 T(2) = [0] because of no solution and n < 4. %e A378488 T(4) = [10, 13] %e A378488 0 1 2 3 0 1 2 3 %e A378488 +---------+ +---------+ %e A378488 0 | . Q . . | | . . Q . | %e A378488 1 | . . . Q | | Q . . . | %e A378488 2 | Q . . . | | . . . Q | %e A378488 3 | . . Q . | | . Q . . | %e A378488 +---------+ +---------+ %e A378488 Permutation: Permutation: %e A378488 [1, 3, 0, 2] [2, 0, 3, 1] %e A378488 Rank: 10 Rank: 13 %e A378488 T(6) = [186, 346, 373, 533]: %e A378488 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 %e A378488 +-------------+ +-------------+ +-------------+ +------------- %e A378488 0 | . Q . . . . | | . . Q . . . | | . . . Q . . | | . . . . Q . | %e A378488 1 | . . . Q . . | | . . . . . Q | | Q . . . . . | | . . Q . . . | %e A378488 2 | . . . . . Q | | . Q . . . . | | . . . . Q . | | Q . . . . . | %e A378488 3 | Q . . . . . | | . . . . Q . | | . Q . . . . | | . . . . . Q | %e A378488 4 | . . Q . . . | | Q . . . . . | | . . . . . Q | | . . . Q . . | %e A378488 5 | . . . . Q . | | . . . Q . . | | . . Q . . . | | . Q . . . . | %e A378488 +-------------+ +-------------+ +-------------+ +-------------+ %e A378488 Permutation: Permutation: Permutation: Permutation: %e A378488 [1, 3, 5, 0, 2, 4] [2, 5, 1, 4, 0, 3] [3, 0, 4, 1, 5, 2] [4, 2, 0, 5, 3, 1] %e A378488 Rank:186 Rank:346 Rank: 373 Rank: 533 %o A378488 (Python) %o A378488 from sympy.combinatorics import Permutation %o A378488 def queens(n, i = 0, cols=0, pos_diags=0, neg_diags=0, sol=None): %o A378488 if sol is None: sol = [] %o A378488 if i == n: yield sol[:] %o A378488 else: %o A378488 neg_diag_mask_ = 1 << (i+n) %o A378488 col_mask = 1 %o A378488 for j in range(n): %o A378488 col_mask <<= 1 %o A378488 pos_diag_mask = col_mask << i %o A378488 neg_diag_mask = neg_diag_mask_ >> (j+1) %o A378488 if not (cols & col_mask or pos_diags & pos_diag_mask or neg_diags & %o A378488 neg_diag_mask): %o A378488 sol.append(j) %o A378488 yield from queens(n, i + 1, %o A378488 cols | col_mask, %o A378488 pos_diags | pos_diag_mask, %o A378488 neg_diags | neg_diag_mask, %o A378488 sol) %o A378488 sol.pop() %o A378488 row = lambda n: [Permutation(sol).rank() for sol in queens(n)] if n >= 4 else [0] %Y A378488 Cf. A000170. %K A378488 nonn,tabf %O A378488 1,4 %A A378488 _Darío Clavijo_, Nov 28 2024