A329235
Number of nonequivalent symmetric sets whose translations cover {1..n}.
Original entry on oeis.org
1, 2, 3, 5, 6, 10, 12, 19, 23, 36, 44, 68, 84, 128, 161, 243, 308, 462, 592, 882, 1140, 1690, 2200, 3249, 4255, 6264, 8246, 12110, 16008, 23466, 31128, 45566, 60618, 88644, 118205, 172731, 230782, 337072, 451082, 658628, 882582, 1288432, 1728484, 2523104, 3388084
Offset: 1
For n = 6 there are 10 symmetric sets (up to equivalence) that with their translations cover {1..6}:
{{1}, {2}, {3}, {4}, {5}, {6}};
{{1, 4}, {2, 5}, {3, 6}};
{{1, 3}, {2, 4}, {3, 5}, {4, 6}};
{{1, 3, 5}, {2, 4, 6}};
{{1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6}};
{{1, 2, 4, 5}, {2, 3, 5, 6}};
{{1, 2, 3}, {2, 3, 4}, {3, 4, 5}, {4, 5, 6}};
{{1, 2, 3, 4}, {2, 3, 4, 5}, {3, 4, 5, 6}};
{{1, 2, 3, 4, 5}, {2, 3, 4, 5, 6}};
{{1, 2, 3, 4, 5, 6}}.
Cf.
A079500 (if symmetry is not required).
A356114
Number of irreducible permutations of n with partition type [2, 1, 1, ..., 1] (with '1' taken n - 2 times).
Original entry on oeis.org
0, 0, 0, 2, 9, 24, 55, 118, 245, 500, 1011, 2034, 4081, 8176, 16367, 32750, 65517, 131052, 262123, 524266, 1048553, 2097128, 4194279, 8388582, 16777189, 33554404, 67108835, 134217698, 268435425, 536870880, 1073741791, 2147483614, 4294967261, 8589934556, 17179869147
Offset: 0
a(4) = 9 = card({2413, 2431, 3142, 3241, 3421, 4132, 4213, 4231, 4312}). The other two permutations of type [2, 1, 1], 1432 and 3214, are reducible. That there are 11 permutations of type [2, 1, 1] we know from Euler's triangle A173018 or from its refined form A355777.
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seq(`if`(n < 3, 0, combinat:-eulerian1(n, n - 2) - 2), n = 0..34);
A368746
Compositions (ordered partitions) of n into odd parts where the first part must be a maximal part.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 18, 27, 40, 61, 93, 142, 217, 333, 512, 789, 1217, 1881, 2912, 4514, 7007, 10893, 16956, 26427, 41238, 64426, 100767, 157778, 247301, 388007, 609351, 957836, 1506928, 2372763, 3739035, 5896462, 9305388, 14695124, 23221657, 36718116, 58092690, 91961034
Offset: 0
The a(10) = 18 such compositions are:
1: [ 1 1 1 1 1 1 1 1 1 1 ]
2: [ 3 1 1 1 1 1 1 1 ]
3: [ 3 1 1 1 1 3 ]
4: [ 3 1 1 1 3 1 ]
5: [ 3 1 1 3 1 1 ]
6: [ 3 1 3 1 1 1 ]
7: [ 3 1 3 3 ]
8: [ 3 3 1 1 1 1 ]
9: [ 3 3 1 3 ]
10: [ 3 3 3 1 ]
11: [ 5 1 1 1 1 1 ]
12: [ 5 1 1 3 ]
13: [ 5 1 3 1 ]
14: [ 5 3 1 1 ]
15: [ 5 5 ]
16: [ 7 1 1 1 ]
17: [ 7 3 ]
18: [ 9 1 ]
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b:= proc(n, m) option remember; `if`(n=0, 1, `if`(m=0,
add(b(n-2*j+1, 2*j-1), j=1..(n+1)/2), add(
b(n-2*j+1, min(n-2*j+1, m)), j=1..(min(n, m)+1)/2)))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..45); # Alois P. Heinz, Jan 04 2024
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b[n_, m_] := b[n, m] = If[n == 0, 1, If[m == 0,
Sum[b[n - 2j + 1, 2j - 1], {j, 1, (n + 1)/2}], Sum[
b[n - 2j + 1, Min[n - 2j + 1, m]], {j, 1, (Min[n, m] + 1)/2}]]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 03 2024, after Alois P. Heinz *)
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my(N=44, x='x+O('x^N)); Vec(1+sum(n=1, N, x^(2*n-1)/(1-sum(k=1, n, x^(2*k-1)))))
A186523
In lunar arithmetic in base 3, this is the number of lunar divisors of the number 111...1 (with n 1's).
Original entry on oeis.org
2, 6, 14, 34, 82, 206, 526, 1378, 3674, 9950, 27278, 75514, 210634, 591054, 1666406, 4716186, 13389434, 38113462, 108737894, 310851770, 890244546, 2553782246, 7337143110, 21110531858, 60823037522, 175471424214, 506862346750, 1465875933394, 4244315841666, 12302751068958, 35699515448510, 103697897085458, 301514033677482, 877521633678990
Offset: 1
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