cp's OEIS Frontend

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

%I A354528 #26 Sep 23 2022 03:46:56
%S A354528 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,
%T A354528 45,23,31,60,77,87,77,60,31,39,77,99,117,119,117,99,77,39,49,96,127,
%U A354528 151,161,151,127,96,49,59,117,155,189,203,213
%N A354528 Square array T(m,n) read by antidiagonals - see Comments for definition.
%C A354528 T(m,n) is defined as follows:
%C A354528 T(m, n) = T(n, m).
%C A354528 T(1, n) = floor(n^2/2) - 1.
%C A354528 T(2, n) = (n+1)^2 - 4.
%C A354528 For m, n >= 3 we have:
%C A354528 T(m, n) = m*n*(m + n)/2 - 3,             if m and n are both even;
%C A354528         = m*n*(m + n)/2 - (m + n)/2 - 1, if m and n are both odd;
%C A354528         = m*n*(m + n)/2 - n/2 - 1,       if m is odd and n is even.
%C A354528 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.
%C A354528 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.
%C A354528 The conjecture is proved if m = 1 or n = 1.
%D A354528 L. Bulteau, S. Giraudo and S. Vialette, Disorders and Permutations, CPM, 2021.
%H A354528 Sela Fried, <a href="https://arxiv.org/abs/2208.03788">The disorder number of a graph</a>, arXiv:2208.03788 [math.CO], 2022.
%e A354528 m\n   1  2  3   4   5   6 ...
%e A354528 1     0  1  3   7  11  17
%e A354528 2     1  5 12  21  32  45
%e A354528 3     3 12 23  39  55  77
%e A354528 4     7 21 39  61  87 117
%e A354528 5    11 32 55  87 119 161
%e A354528 6    17 45 77 117 161 213
%e A354528 ...
%Y A354528 Cf. A047838, A028347, A179094, A354529.
%K A354528 nonn,tabl
%O A354528 1,4
%A A354528 _Sela Fried_, Aug 16 2022