A354529 -id:A354529 - cp's OEIS frontend

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

Sela Fried, Aug 16 2022

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.

			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
...

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

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

Cf. A047838, A028347, A179094, A354529.

cp's OEIS Frontend

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

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