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%I A381896 #39 Mar 25 2025 21:10:56
%S A381896 1,2,6,41
%N A381896 Number of n X n Erdős matrices up to equivalence.
%C A381896 An n X n bistochastic matrix A is Erdős if Sum_{i=1..n} Sum_{j=1..n} A(i, j)^2 = max_{p in S_n} Sum_{i=1..n} A(i, p(i)), where S_n is the set of all permutations of [n]. It is known that any bistochastic matrix satisfies the inequality Sum_{i=1..n} Sum_{j=1..n} A(i, j)^2 <= max_{p in S_n} Sum_{i=1..n} A(i, p(i)).
%C A381896 Two Erdős matrices A and B are equivalent if A=PBQ for some permutation matrices P and Q.
%C A381896 The n-th term in the sequence a(n) >= p(n), where p(n) is the number of partitions of n [Tripathi, 2024].
%H A381896 Ludovick Bouthat, Javad Mashreghi, and Frédéric Morneau-Guérin, <a href="https://doi.org/10.1080/03081087.2023.2300674">On a question of Erdős on doubly stochastic matrices</a>, Linear and Multilinear Algebra, Vol 72, (17) 2024, 2823-2844; <a href="https://arxiv.org/abs/2306.05518">arXiv preprint</a> arXiv:2306.05518 [math.MG], 2023.
%H A381896 Aman Kushwaha and Raghavendra Tripathi, <a href="https://arxiv.org/abs/2503.09542">A note on Erdős matrices and Marcus--Ree inequality</a>, arXiv:2503.09542 [math.MG], 2025.
%H A381896 Raghavendra Tripathi, <a href="https://doi.org/10.1016/j.laa.2024.12.002">Some observations on Erdős matrices</a>, Linear Algebra and its Applications, Vol 708, 2025, 236-251; <a href="https://arxiv.org/abs/2410.06612">arXiv preprint</a> arXiv:2410.06612 [math.MG], 2024.
%e A381896 When n=1, there is only one bistochastic matrix, namely [1], which is clearly Erdős. This gives a(1)=1.
%e A381896 When n=2, there are only 3 Erdős matrices, namely [1, 0; 0, 1], [1/2, 1/2; 1/2, 1/2], and [0, 1; 1, 0]. Since [1, 0; 0, 1] and [0, 1; 1, 0] are equivalent, it follows that a(2)=2.
%e A381896 When n=3, there are only 6 Erdős matrices up to equivalence. This was shown by Bouthat, Mashreghi, and Morneau-Guérin in 2024. Here is a list of 6 non-equivalent Erdős matrices in dimension 3:
%e A381896 [1, 0, 0; 0, 1, 0; 0, 0, 1],
%e A381896 [1/3, 1/3, 1/3; 1/3, 1/3, 1/3; 1/3, 1/3, 1/3],
%e A381896 [1, 0, 0 ; 0, 1/2, 1/2; 0, 1/2, 1/2],
%e A381896 [0 , 1/2, 1/2; 1/2, 1/4, 1/4; 1/2, 1/4, 1/4],
%e A381896 [0, 1/2, 1/2; 1/2, 0, 1/2; 1/2, 1/2, 0],
%e A381896 [3/5, 0, 2/5; 0, 3/5, 2/5; 2/5, 2/5, 1/5]
%e A381896 A complete list 41 of non-equivalent Erdős matrices in dimension 4 was obtained by Kushwaha and Tripathi.
%Y A381896 Cf. A000041.
%K A381896 nonn,hard,more
%O A381896 1,2
%A A381896 _Raghavendra Tripathi_, Mar 09 2025