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A306235 Indices in A306428 of permutations t with a finite number of nonfixed points and such that t_i - t_j <> j - i for any distinct i and j (see Comments for precise definition).

%I A306235 #66 Mar 05 2022 23:01:23
%S A306235 0,2,4,7,8,14,15,24,28,32,33,39,48,56,60,63,64,72,80,87,96,104,111,
%T A306235 121,122,127,134,135,138,140,142,147,150,156,159,160,168,176,184,185,
%U A306235 192,202,207,242,246,247,258,277,296,312,314,316,318,322,326,327,333,366,367,385,414,415,416,420,423,426,428,432,438,443,447,504,505,506,536,537,540,567,569,602,604,628,660
%N A306235 Indices in A306428 of permutations t with a finite number of nonfixed points and such that t_i - t_j <> j - i for any distinct i and j (see Comments for precise definition).
%C A306235 Let T be the set of permutations of nonnegative integers t such that t_i = i for all but a finite number of terms i.
%C A306235 The A306428 sequence enumerates the elements of T, hence we have a bijection f from T to the nonnegative integers.
%C A306235 The bijection f has the following properties: for any N > 0:
%C A306235 - if f(t) < N!, then t_i = i for any i >= N,
%C A306235 - this is consistent with the fact that there are N! permutations of (0..N-1),
%C A306235 - if f(t) + f(u) = N!-1, then t_i = u_{N-1-i} for i = 0..N-1,
%C A306235 - in other words, t and u, restricted to (0..N-1), are symmetrical permutations.
%C A306235 This sequence corresponds to the values f(t) of the permutations t in T such that t_i - t_j <> j - i for any distinct i and j.
%C A306235 Hence, for any n > 0 and N > 0:
%C A306235 - if a(n) < N!, then a(n) represents a permutation t of (0..N-1) such that the numbers t_i + i are distinct for i = 0..N-1; this corresponds to a configuration of N queens on an N X N board in which two queens do not attack each other if they are on the same northwest-southeast diagonal,
%C A306235 - this explains the expression of A099152 in the Formula section,
%C A306235 - also if a(n) = N! - 1 - a(m) for some m > 0, then a(n) represents a permutation t of (0..N-1) such that the numbers t_i + i are distinct for i = 0..N-1 and the numbers t_j - j are distinct for j = 0..N-1; this corresponds to a configuration of N nonattacking queens on an N X N board,
%C A306235 - this explains the expression of A000170 in the Formula section.
%H A306235 Wikipedia, <a href="https://en.wikipedia.org/wiki/Factorial_number_system">Factorial number system</a>
%F A306235 A099152(k) = Sum_{i > 0} [k! - 1 - a(i) >= 0] (with [] = Iverson bracket).
%F A306235 A000170(k) = Sum_{i > 0} [k! - 1 - a(i) belongs to {a(n)}].
%e A306235 For N = 6, there are 83 matrices in which the sums of the entries of each northeast-southwest diagonal are 0 or 1.
%e A306235 Also, for N = 6, there are 4 ways to place 6 nonattacking queens on a 6 X 6 board.
%e A306235 Finally, the solutions for N = 6 are 150, 296, 423 and 569 (positions within the ordered permutations, see A306428).
%e A306235 150 = (2,4,6,1,3,5);
%e A306235 O O O X O O
%e A306235 X O O O O O
%e A306235 O O O O X O
%e A306235 O X O O O O
%e A306235 O O O O O X
%e A306235 O O X O O O
%e A306235 296 = (3,6,2,5,1,4);
%e A306235 O O O O X O
%e A306235 O O X O O O
%e A306235 X O O O O O
%e A306235 O O O O O X
%e A306235 O O O X O O
%e A306235 O X O O O O
%e A306235 423 = (4,1,5,2,6,3);
%e A306235 O X O O O O
%e A306235 O O O X O O
%e A306235 O O O O O X
%e A306235 X O O O O O
%e A306235 O O X O O O
%e A306235 O O O O X O
%e A306235 569 = (5,3,1,6,4,2);
%e A306235 O O X O O O
%e A306235 O O O O O X
%e A306235 O X O O O O
%e A306235 O O O O X O
%e A306235 X O O O O O
%e A306235 O O O X O O
%Y A306235 Cf. A000170, A099152, A306428
%K A306235 nonn
%O A306235 1,2
%A A306235 _Raúl Mario Torres Silva_, Jan 30 2019