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A231428 Sorted and encoded binary matrices representing equivalence relations.

%I A231428 #37 Mar 13 2025 04:24:46
%S A231428 0,1,2,4,7,8,12,16,18,25,32,33,42,52,63,64,68,80,96,116,128,130,136,
%T A231428 160,170,193,225,256,257,264,272,281,322,338,388,396,455,512,513,514,
%U A231428 516,519,584,588,656,658,729,800,801,874,948,1023,1024
%N A231428 Sorted and encoded binary matrices representing equivalence relations.
%C A231428 The N X N binary matrix of an equivalence relation is perfectly defined by its upper-right triangle. We encode such a matrix with the (N*(N-1))/2 bit number obtained by joining together each line of the upper-right triangle. The numbers are converted to base 10.
%C A231428 This is an infinite sequence and can be used for arbitrarily large values of N. To enumerate the finite list of n X n matrices for a given n, truncate this sequence to the first A000110(n) elements.
%H A231428 Philippe Beaudoin, <a href="/A231428/b231428.txt">Table of n, a(n) for n = 1..10000</a>
%H A231428 Philippe Beaudoin, <a href="/A231428/a231428_1.py.txt">Python program to generate the sequence</a>
%H A231428 Tilman Piesk, <a href="https://commons.wikimedia.org/wiki/File:Set_partitions_5;_list.svg">Illustrated list of the first 52 equivalence relations</a>
%H A231428 Tilman Piesk, <a href="/A231428/a231428.txt">Non-singleton blocks of the partitions of {1...8} in the same order.</a>
%H A231428 Tilman Piesk, <a href="https://en.wikiversity.org/wiki/Permutations_and_partitions_in_the_OEIS">Permutations and partitions in the OEIS</a> (Wikiversity)
%e A231428 The 4 X 4 equivalence matrices are represented by the first A000110(4) = 15 elements, that is: 0, 1, 2, 4, 7, 8, 12, 16, 18, 25, 32, 33, 42, 52, 63.
%e A231428 The 4 X 4 matrix corresponding to 25 = 0b011001 is:
%e A231428   1 0 1 1
%e A231428   0 1 0 0
%e A231428   1 0 1 1
%e A231428   1 0 1 1
%K A231428 nonn,base
%O A231428 1,3
%A A231428 _Philippe Beaudoin_, Nov 09 2013