A076831 -id:A076831 - cp's OEIS frontend

A227960 Big equivalence classes (A227723) related to subgroups of nimber addition (A190939).

1, 3, 6, 15, 24, 60, 105, 255, 384, 960, 1632, 1680, 4080, 15555, 27030, 65535, 98304, 245760, 417792, 430080, 1044480, 1582080, 3947520, 3982080, 6908160, 6919680, 16776960, 106991625, 267448335, 1019462460, 1771476585, 4294967295

Offset: 0

Tilman Piesk, Aug 01 2013

A subsequence of A227723, showing all the big equivalence classes that contain Boolean functions related to subgroups of nimber addition (A190939).
Forms a triangle with row lengths A034343 = 1, 1, 2, 4, 8, 16, 36, 80...:
1,
3,
6, 15,
24, 60, 105, 255,
384, 960, 1632, 1680, 4080, 15555, 27030, 65535...
The left column a( 1,2,4,8,16,32,68,148... ) = a( A076766 ) = 3 ,6, 24, 384, 98304... is probably A001146 * 3/2, which is also A006017( A000079 ).
The first A076766(n) entries correspond to the first A006116(n) entries of A190939. (The first 148 here, for n = 7,  correspond to the first 29212 there.) The entries of A190939 can be generated from this sequence.
Among the first A076766(n) entries are A076831(n;0...n) with weight 2^0...2^n. (Among the first 148 are 1, 7, 23, 43, 43, 23, 7, 1 with weights 1, 2, 4, 8, 16, 32, 64, 128.)
a(n) appears to be divisible by 3 for n>0, and the odd part of a(n) is almost always squarefree. - Ralf Stephan, Aug 02 2013

Tilman Piesk, Table of n, a(n) for n = 0..147
Tilman Piesk, a(n) for n = 0..147 and corresponding entries of A190939 in reverse binary. Prime factors and numbers of prime factors (A001222).
Tilman Piesk, Subgroups of nimber addition (Wikiversity)

Subsequence of A227723 (all becs). All entries are also in A227963 (all sona-secs). Neither shares the property of divisibility by 3.
A190939, A034343, A076766, A076831.
The prime factors contain many prime factors of Fermat numbers (A023394).

a( A076766 - 1 ) = A001146 - 1 = A051179.

a( A076766 ) = A001146 * 3/2 (probably).

A347970 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_3)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 8, 16, 8, 1, 1, 11, 39, 39, 11, 1, 1, 15, 87, 168, 87, 15, 1, 1, 19, 176, 644, 644, 176, 19, 1, 1, 24, 338, 2348, 4849, 2348, 338, 24, 1, 1, 29, 613, 8137, 37159, 37159, 8137, 613, 29, 1, 1, 35, 1071, 27047, 286747, 679054, 286747, 27047, 1071

Offset: 0

Álvar Ibeas, Sep 21 2021

Columns can be computed by a method analogous to that of Fripertinger for isometry classes of linear codes, disallowing scalar transformation of individual coordinates.

			Triangle begins:
  k:  0   1   2   3   4   5   6   7
      -----------------------------
n=0:  1
n=1:  1   1
n=2:  1   3   1
n=3:  1   5   5   1
n=4:  1   8  16   8   1
n=5:  1  11  39  39  11   1
n=6:  1  15  87 168  87  15   1
n=7:  1  19 176 644 644 176  19   1
There are 4 = A022167(2, 1) one-dimensional subspaces in (F_3)^2, namely, those generated by (0, 1), (1, 0), (1, 1), and (1, 2). The first two are related by coordinate swap, while the remaining two are invariant. Hence, T(2, 1) = 3.

Álvar Ibeas, Entries up to T(16, 7)
H. Fripertinger, Isometry classes of codes
H. Fripertinger, Number of the isometry classes of all ternary (n,k)-codes
Álvar Ibeas, Column k=2 up to n=100
Álvar Ibeas, Column k=3 up to n=100
Álvar Ibeas, Column k=4 up to n=100
Álvar Ibeas, Column k=5 up to n=100
Álvar Ibeas, Column k=6 up to n=100
Álvar Ibeas, Column k=7 up to n=100

Cf. A022167, A024206(n+1) (column k=1), A076831.

A348113 Irregular triangle read by rows: T(n, k) is the number of chains of subspaces 0 < V_1 < ... < V_r = (F_2)^n, counted up to coordinate permutation, with dimension increments given by (any fixed permutation of) the parts of the k-th partition of n in Abramowitz-Stegun order.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 6, 13, 28, 1, 5, 10, 23, 37, 85, 196, 1, 6, 16, 22, 37, 87, 149, 207, 357, 864, 2109, 1, 7, 23, 43, 55, 180, 269, 479, 441, 1193, 2169, 2992, 5483, 13958, 35773, 1, 8, 32, 77, 106, 78, 341, 734, 1354, 2153, 856, 3468, 5559, 10544, 20185, 8943, 27572, 53115, 72517, 140563, 373927

Offset: 1

Álvar Ibeas, Oct 01 2021

A permutation on the list of dimension increments does not modify the number of subspace chains.

The length of the enumerated chains is r = len(L), where L is the parameter partition.

			For L = (1, 1, 1), there are 21 (= 7 * 3) = A347485(3, 3) subspace chains 0 < V_1 < V_2 < (F_2)^3.
The permutations of the three coordinates classify them into 6 = T(3, 3) orbits:
<e_1>, <e_1, e_2>;                     <e_1>, <e_1, e_2 + e_3>;
<e_1 + e_2>, <e_1, e_2>;               <e_1 + e_2>, <e_1 + e_2, e_3>;
<e_1 + e_2>, <e_1 + e_2, e_1 + e_3>;   <e_1 + e_2 + e_3>, <e_1 + e_2, e_3>.
T(3, 2) = 3 refers to partition (2, 1) and counts subspace chains in (F_2)^2 with dimensions (0, 2, 3), i.e., 2-dimensional subspaces. It also counts chains with dimensions (0, 1, 3), i.e., 1-dimensional subspaces.
Triangle begins:
  k:  1 2  3  4  5  6   7   8   9  10   11
      ------------------------------------
n=1:  1
n=2:  1 2
n=3:  1 3  6
n=4:  1 4  6 13 28
n=5:  1 5 10 23 37 85 196
n=6:  1 6 16 22 37 87 149 207 357 864 2109

Álvar Ibeas, Table of n, a(n) for n = 1..137
Álvar Ibeas, First 16 rows, with gaps
Álvar Ibeas, Pseudo-column T(n, L), where L = (n-2, 1, 1), up to n=100
Álvar Ibeas, Pseudo-column T(n, L), where L = (n-3, 2, 1), up to n=100
Álvar Ibeas, Pseudo-column T(n, L), where L = (n-3, 1, 1, 1), up to n=100
Álvar Ibeas, Pseudo-column T(n, L), where L = (n-4, 3, 1), up to n=100
Álvar Ibeas, Pseudo-column T(n, L), where L = (n-4, 2, 2), up to n=100
Álvar Ibeas, Pseudo-column T(n, L), where L = (n-4, 2, 1, 1), up to n=100
Álvar Ibeas, Pseudo-column T(n, L), where L = (n-4, 1, 1, 1, 1), up to n=100

Cf. A076831, A347485.

If the k-th partition of n in A-St is L = (a, n-a), then T(n, k) = A076831(n, a) = A076831(n, n-a).

A250002 Triangle read by rows: T(n,k) = number of inequivalent binary linear [n,k] codes minus C(n,k).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 2, 8, 8, 2, 0, 0, 0, 0, 4, 21, 36, 21, 4, 0, 0, 0, 0, 7, 47, 114, 114, 47, 7, 0, 0, 0, 0, 11, 93, 306, 453, 306, 93, 11, 0, 0, 0, 0, 16, 168, 730, 1526, 1526, 730, 168, 16, 0, 0

Offset: 0

Tilman Piesk, Nov 10 2014

The triangle of inequivalent binary linear [n,k] codes (A076831) looks much like Pascal's triangle (A007318). They start to differ in the middle of row 6. This triangle is the difference between them. Its row sums are A250003 - the difference between the numbers of inequivalent binary linear codes of length n (A076766) and the powers of two (A000079).

			      k   0   1   2   3   4    5    6   7   8   9  10  11      sums
   n
   0      0                                                       0
   1      0   0                                                   0
   2      0   0   0                                               0
   3      0   0   0   0                                           0
   4      0   0   0   0   0                                       0
   5      0   0   0   0   0    0                                  0
   6      0   0   1   2   1    0    0                             4
   7      0   0   2   8   8    2    0   0                        20
   8      0   0   4  21  36   21    4   0   0                    86
   9      0   0   7  47 114  114   47   7   0   0               336
  10      0   0  11  93 306  453  306  93  11   0   0          1273
  11      0   0  16 168 730 1526 1526 730 168  16   0   0      4880
Row 6 of A076831 is (1,6,16,22,16,6,1) and row 6 of A007318 is (1,6,15,20,15,6,1). Row 6 of this triangle is their difference (0,0,1,2,1,0,0).

Cf. A076831, A007318, A250003.

a(n,k) = A076831(n,k) - A007318(n,k).

A174743 Partial sums of A076766.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 131, 279, 621, 1469, 3766, 10694, 34728, 133582, 636719, 3955451, 33664265, 407703531, 7147333784, 180948983492, 6537204164708, 332740721681412, 23627093701822296, 2324803141466748032, 315610211340647518667, 58953876234603150481383

Offset: 0

Jonathan Vos Post, Nov 30 2010

nonn

Number of inequivalent binary linear codes of length <= n. Also the total number of nonisomorphic binary matroids on an k-set for all k <= n. The subsequence of primes is: 3, 7, 31, 131.

			a(14) = 1 + 2 + 4 + 8 + 16 + 32 + 68 + 148 + 342 + 848 + 2297 + 6928 + 24034 + 98854 + 503137 = 636719 is prime.

Cf. A076766, A076831, A034328, A055545.

cp's OEIS Frontend

A227960 Big equivalence classes (A227723) related to subgroups of nimber addition (A190939).

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A347970 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_3)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

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A348113 Irregular triangle read by rows: T(n, k) is the number of chains of subspaces 0 < V_1 < ... < V_r = (F_2)^n, counted up to coordinate permutation, with dimension increments given by (any fixed permutation of) the parts of the k-th partition of n in Abramowitz-Stegun order.

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A250002 Triangle read by rows: T(n,k) = number of inequivalent binary linear [n,k] codes minus C(n,k).

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A174743 Partial sums of A076766.

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