A385176 Positive half of inverse speed permutation array. Square array A(n,k), n >= 0, k >= 0, read by ascending antidiagonals.

1, -1, 2, 2, 2, 3, 2, -1, 3, 4, -2, 3, 3, 4, 5, -2, 3, -1, 4, 5, 6, 3, 3, 4, 4, 5, 6, 7, 3, -2, 4, -1, 5, 6, 7, 8, 3, -2, 4, 5, 5, 6, 7, 8, 9, -3, 4, 4, 5, -1, 6, 7, 8, 9, 10, -3, 4, -2, 5, 6, 6, 7, 8, 9, 10, 11, -3, 4, -2, 5, 6, -1, 7, 8, 9, 10, 11, 12, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13

Offset: 0

Author

Peter Munn, Jun 20 2025

Keywords

• sign
• tabl

Comments

Particles labeled with nonzero integers j start at time t = 0 at x = 2k (offset from the origin) on a straight line. Each particle, j, moves at speed -1/j, so crosses the origin at time t = 2j^2. T(n,k) gives the label of the particle in the line segment (2k, 2k+2) at time t = 2n+1.

It is easy to determine that particles labeled i and -j cross at x = 2*(i-j) at time t = 2ij, and that (for t > 0) a particle crosses x = 2k only when encountering a particle heading in the opposite direction. So at t = 2n+1 there is exactly one particle in each segment (2k, 2k+2) and the particle labels define a bi-infinite permution of the nonzero integers. For the terms of this sequence, we restrict k >= 0; and taking the absolute values of the terms in each row gives a permutation of the positive integers. Moreover, the differences between row n-1 and row n consist of exchanges of paired divisors of -n.

The halved positions, k, at which particles encounter a segment boundary x = 2k at t = 2n are given by row n of A368312. So when that row starts with a 0, this indicates a particle crossing the origin. On the other hand, the nonzero terms, k, of row t of A211343 indicate the segment midpoints x = 2k-1 that are encountered by particles at time t, with terms in odd (respectively even) columns corresponding to positive-labeled (respectively negative-labeled) particles.

Examples

			Square array A(n,k) begins:
   n   t\k|   0    1    2    3    4    5    6    7    8    9   10   11   12
  --------+-----------------------------------------------------------------
   0   1  |   1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13
   1   3  |  -1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13
   2   5  |   2,  -1,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13
   3   7  |   2,   3,  -1,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13
   4   9  |  -2,   3,   4,  -1,   5,   6,   7,   8,   9,  10,  11,  12,  13
   5  11  |  -2,   3,   4,   5,  -1,   6,   7,   8,   9,  10,  11,  12,  13
   6  13  |   3,  -2,   4,   5,   6,  -1,   7,   8,   9,  10,  11,  12,  13
   7  15  |   3,  -2,   4,   5,   6,   7,  -1,   8,   9,  10,  11,  12,  13
   8  17  |   3,   4,  -2,   5,   6,   7,   8,  -1,   9,  10,  11,  12,  13
   9  19  |  -3,   4,  -2,   5,   6,   7,   8,   9,  -1,  10,  11,  12,  13
  10  21  |  -3,   4,   5,  -2,   6,   7,   8,   9,  10,  -1,  11,  12,  13
  11  23  |  -3,   4,   5,  -2,   6,   7,   8,   9,  10,  11,  -1,  12,  13
  12  25  |   4,  -3,   5,   6,  -2,   7,   8,   9,  10,  11,  12,  -1,  13
		

Crossrefs

Cf. A211343, A368312.