cp's OEIS Frontend

A378609 Number of weak orderings of (i,j), 1 <= i,j <= n, for which there is a set X = {x_1<... of n real numbers with (i,j) < (k,m) if and only if x_i + x_j < x_k + x_m.

%I A378609 #18 Dec 16 2024 14:49:58
%S A378609 1,1,3,25,477,18403
%N A378609 Number of weak orderings of (i,j), 1 <= i,j <= n, for which there is a set X = {x_1<...<x_n} of n real numbers with (i,j) < (k,m) if and only if x_i + x_j < x_k + x_m.
%C A378609 Number of weak orderings of (i,j), 1<=i,j<=n, for which there is a set X={x_1<...<x_n} of n integers with (i,j)<(k,m) if and only if x_1+x_j < x_k + x_m.
%C A378609 Number of weak orderings of (i,j), 1<=i,j<=n, for which there is a set X={x_1<...<x_n} of n positive integers with (i,j)<(k,m) if and only if x_1 * x_j < x_k * x_m.
%H A378609 Kevin O'Bryant, <a href="https://arxiv.org/abs/2411.08139">Visualizing the sum-product conjecture</a>, arXiv:2411.08139 [math.NT], 2024.
%F A378609 a(n) <= A376162(n).
%e A378609 For n=3 the three weak orderings are: 11<12=21<13=31=22<23=32<33, 11<12=21<22<13=31<23=32<33, and 11<12=21<13=31<22<23=32<33, which arise from the sets {1,2,3}, {1,2,4}, and {1,3,4}, respectively.
%e A378609 For n=4, there are 39 weak orderings, but only 25 arise from sets of real numbers. For example, the weak ordering 11<12=21<13=31<22<14=41=23=32<24=42=33<34=43<44 does not arise from any set of real numbers.
%Y A378609 Cf. A376162.
%K A378609 nonn,hard,more
%O A378609 1,3
%A A378609 _Kevin O'Bryant_, Dec 01 2024