This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A362895 #35 Jul 23 2025 16:04:52
%S A362895 1,1,1,1,1,3,3,1,1,4,12,12,6,6,4,1,1,5,20,30,30,60,60,20,10,10,30,30,
%T A362895 10,10,5,1,1,6,30,60,60,180,180,60,60,120,360,360,180,180,120,30,15,
%U A362895 15,60,90,90,180,180,60,20,20,60,60,15,15,6,1,1,7
%N A362895 a(n) is the length of the smallest orbit of the n-th natural downset.
%C A362895 Under a partial order based on the bitwise OR operation (see Wikipedia link) as a join, the set N_n = {0,1,2,...,n-1} is downward closed for all nonnegative integers n. Let N_n under the bitwise ordering be the n-th natural downset. E.g., for n = 0 to n = 9, the sets N_0 through N_9 under a bitwise ordering form the downward closed posets represented by the following Hasse diagrams:
%C A362895 7 7
%C A362895 / | \ / | \
%C A362895 3 3 3 5 3 5 6 3 5 6 3 5 6
%C A362895 / \ | \ | X \ | X X | | X X | | X X |
%C A362895 1 1 2 1 2 1 2 4 1 2 4 1 2 4 1 2 4 1 2 4 8
%C A362895 | \ / \ / \ | / \ | / \ | / \ | / \ \ / /
%C A362895 {} 0 0 0 0 0 0 0 0 0
%C A362895 The sequence {a(n)} lists the length of the orbit of the n-th natural downset under permutations of its atoms. Equivalently, given the smallest number k such that n <= 2^k, a(n) is the number of labeled downsets of the Boolean lattice of size 2^k which are isomorphic to the n-th natural downset (see examples for an illustration of n = 5).
%C A362895 Intuitively, this can be understood as the number of ways to internally rotate the n-th natural downset within this smallest Boolean lattice that it can fit while still being a downset.
%C A362895 More generally, for any nonnegative integer m, the number of labeled downsets of the Boolean lattice of size 2^m which are isomorphic to the n-th natural downset is given by a(n)*binomial(m,k). Thus, a(n) is the smallest (nonzero) orbit length, which obtains when binomial(m,k) = 1.
%C A362895 Note that k is the number of elements in the 1st rank of the posets, which is also the number of vertices in the corresponding simplicial complex, or k = ceiling(log_2(n)) for n > 0.
%H A362895 Bruno L. O. Andreotti, <a href="/A362895/b362895.txt">Table of n, a(n) for n = 0..9999</a>
%H A362895 Bruno L. O. Andreotti, <a href="/A362895/a362895.py.txt">Python program for n = 0 to 128</a>
%H A362895 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bitwise_operation#OR">Bitwise OR</a>
%F A362895 Let k(n) = ceiling(log_2(n)) for n > 0, j = 2^k(n)-n, and k(j) = ceiling(log_2(j)) if j > 0, or k(j) = 0 if j = 0. Provably, a(n) = a(j)*binomial(k(n),k(j)).
%e A362895 For any nonnegative integer m the natural downset corresponding to N_2^m = {0,1,2,...,(2^m)-1} is a Boolean lattice. For n = 5 we have k = 3 which corresponds to the Boolean lattice N_2^k = N_8. We can illustrate a(5) = 3 under this definition based on the three downsets of N_8 which are isomorphic to N_5 (including N_5 itself):
%e A362895 7
%e A362895 / | \
%e A362895 3 5 6 3 5 6
%e A362895 | X X | : | \ , / \ , / |
%e A362895 1 2 4 1 2 4 1 2 4 1 2 4
%e A362895 \ | / \ | / \ | / \ | /
%e A362895 0 0 0 0
%e A362895 Other examples:
%e A362895 a(0) = 1: N_0 = {} -> {}
%e A362895 a(1) = 1: N_1 = {0} -> {0}
%e A362895 a(2) = 1: N_2 = {0,1} -> {0,1}
%e A362895 a(3) = 1: N_3 = {0,1,2} -> {0,1,2}
%e A362895 a(4) = 1: N_4 = {0,1,2,3} -> {0,1,2,3}
%e A362895 a(5) = 3: N_5 = {0,1,2,3,4} -> {0,1,2,3,4}, {0,1,2,4,5}, {0,1,2,4,6}
%e A362895 a(6) = 3: N_6 = {0,1,2,3,4,5} -> {0,1,2,3,4,5}, {0,1,2,3,4,6}, {0,1,2,4,5,6}
%e A362895 a(7) = 1: N_7 = {0,1,2,3,4,5,6} -> {0,1,2,3,4,5,6}
%e A362895 a(8) = 1: N_8 = {0,1,2,3,4,5,6,7} -> {0,1,2,3,4,5,6,7}
%e A362895 a(9) = 4: N_9 = {0,1,2,3,4,5,6,7,8} -> {0,1,2,3,4,5,6,7,8}, {0,1,2,3,8,9,10,11}, {0,1,4,5,8,9,12,13}, {0,2,4,6,8,10,12,14}
%o A362895 (Python) # See Andreotti link.
%Y A362895 The cardinality of the downset lattice of the n-th natural downset is A132581. When n is a power of 2, the cardinality of the downset lattice of the n-th natural downset is the log_2(n)-th Dedekind number (A000372).
%K A362895 nonn,easy
%O A362895 0,6
%A A362895 _Bruno L. O. Andreotti_, May 09 2023