A354528 Square array T(m,n) read by antidiagonals - see Comments for definition.

0, 1, 1, 3, 5, 3, 7, 12, 12, 7, 11, 21, 23, 21, 11, 17, 32, 39, 32, 17, 23, 45, 55, 61, 55, 45, 23, 31, 60, 77, 87, 77, 60, 31, 39, 77, 99, 117, 119, 117, 99, 77, 39, 49, 96, 127, 151, 161, 151, 127, 96, 49, 59, 117, 155, 189, 203, 213

Offset

1,4

Comments

T(m,n) is defined as follows:

T(m, n) = T(n, m).

T(1, n) = floor(n^2/2) - 1.

T(2, n) = (n+1)^2 - 4.

For m, n >= 3 we have:

T(m, n) = m*n*(m + n)/2 - 3, if m and n are both even;

= m*n*(m + n)/2 - (m + n)/2 - 1, if m and n are both odd;

= m*n*(m + n)/2 - n/2 - 1, if m is odd and n is even.

The disorder number M(G) of a graph G is defined to be the maximal length of a walk along the edges of the graph, according to any ordering of its vertices.

Conjecture: T(m, n) = M(P_m X P_n) where P_m X P_n is the grid graph of size m X n.

The conjecture is proved if m = 1 or n = 1.

References

L. Bulteau, S. Giraudo and S. Vialette, Disorders and Permutations, CPM, 2021.

Links

Table of n, a(n) for n=1..58.

Sela Fried, The disorder number of a graph, arXiv:2208.03788 [math.CO], 2022.

Example

m\n   1  2  3   4   5   6 ...
1     0  1  3   7  11  17
2     1  5 12  21  32  45
3     3 12 23  39  55  77
4     7 21 39  61  87 117
5    11 32 55  87 119 161
6    17 45 77 117 161 213
...

Crossrefs

Cf. A047838, A028347, A179094, A354529.

Sequence in context: A248511 A210191 A110465 * A328915 A100338 A094444

Adjacent sequences: A354525 A354526 A354527 * A354529 A354530 A354531

Keyword

nonn,tabl

Author

Sela Fried, Aug 16 2022

Status

approved