A241926 -id:A241926 - cp's OEIS frontend

A347974 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_8)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

1, 1, 1, 1, 5, 1, 1, 17, 17, 1, 1, 47, 242, 47, 1, 1, 113, 3071, 3071, 113, 1, 1, 245, 34477, 232290, 34477, 245, 1, 1, 491, 341633, 16665755, 16665755, 341633, 491, 1, 1, 920, 3022045, 1073874283, 8241549097, 1073874283, 3022045, 920, 1, 1, 1635, 24145695

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.

Regarding the formula for column k = 1, note that A241926(q - 1, n) counts, up to coordinate permutation, one-dimensional subspaces of (F_q)^n generated by a vector with no zero component.

			Triangle begins:
  k:  0    1    2    3    4    5
      --------------------------
n=0:  1
n=1:  1    1
n=2:  1    5    1
n=3:  1   17   17    1
n=4:  1   47  242   47    1
n=5:  1  113 3071 3071  113    1
There are 9 = A022172(2, 1) one-dimensional subspaces in (F_8)^2. Among them, <(1, 1)> is invariant by coordinate swap and the rest are grouped in orbits of size two. Hence, T(2, 1) = 5.

Álvar Ibeas, Entries up to T(10, 4)
H. Fripertinger, Isometry classes of codes
Álvar Ibeas, Column k=1 up to n=100
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

Cf. A022172, A032192, A241926.

T(n, 1) = T(n - 1, 1) + A032192(n + 7).

A347975 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_9)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 21, 21, 1, 1, 64, 374, 64, 1, 1, 163, 5900, 5900, 163, 1, 1, 380, 82587, 644680, 82587, 380, 1, 1, 809, 1018388, 66136870, 66136870, 1018388, 809, 1, 1, 1619, 11174165, 6057912073, 52901629980, 6057912073, 11174165, 1619, 1, 1, 3049, 110404788

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.

Regarding the formula for column k = 1, note that A241926(q-1, n) counts, up to coordinate permutation, one-dimensional subspaces of (F_q)^n generated by a vector with no zero component.

			Triangle begins:
  k:  0    1    2    3    4    5
      --------------------------
n=0:  1
n=1:  1    1
n=2:  1    6    1
n=3:  1   21   21    1
n=4:  1   64  374   64    1
n=5:  1  163 5900 5900  163    1
There are 10 = A022173(2, 1) one-dimensional subspaces in (F_9)^2. Among them, <(1, 1)> and <(1, 2)> are invariant by coordinate swap and the rest are grouped in orbits of size two. Hence, T(2, 1) = 6.

Álvar Ibeas, Entries up to T(10, 4)
H. Fripertinger, Isometry classes of codes
Álvar Ibeas, Column k=1 up to n=100
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

Cf. A022173, A032193, A241926.

T(n, 1) = T(n-1, 1) + A032193(n+8).

A322596 Square array read by descending antidiagonals (n >= 0, k >= 0): let b(n,k) = (n+k)!/((n+1)!*k!); then T(n,k) = b(n,k) if b(n,k) is an integer, and T(n,k) = floor(b(n,k)) + 1 otherwise.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 4, 3, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 4, 7, 9, 7, 4, 1, 1, 1, 4, 10, 14, 14, 10, 4, 1, 1, 1, 5, 12, 21, 26, 21, 12, 5, 1, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 1, 6, 19, 42, 66, 77, 66, 42, 19, 6, 1, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1, 1

Offset: 0

Franck Maminirina Ramaharo, Jan 22 2019

For n >= 1, T(n,k) is the number of nodes in n-dimensional space for Mysovskikh's cubature formula which is exact for any polynomial of degree k of n variables.

			Array begins:
  1, 1, 1,  1,  1,   1,   1,    1,    1,    1, ...
  1, 1, 2,  2,  3,   3,   4,    4,    5,    5, ...
  1, 1, 2,  4,  5,   7,  10,   12,   15,   19, ...
  1, 1, 3,  5,  9,  14,  21,   30,   42,   55, ...
  1, 1, 3,  7, 14,  26,  42,   66,   99,  143, ...
  1, 1, 4, 10, 21,  42,  77,  132,  215,  334, ...
  1, 1, 4, 12, 30,  66, 132,  246,  429,  715, ...
  1, 1, 5, 15, 42,  99, 215,  429,  805, 1430, ...
  1, 1, 5, 19, 55, 143, 334,  715, 1430, 2702, ...
  1, 1, 6, 22, 72, 201, 501, 1144, 2431, 4862, ...
  ...
As triangular array, this begins:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  1,  1;
  1, 2,  2,  1,  1;
  1, 3,  4,  3,  1,  1;
  1, 3,  5,  5,  3,  1,  1;
  1, 4,  7,  9,  7,  4,  1,  1;
  1, 4, 10, 14, 14, 10,  4,  1, 1;
  1, 5, 12, 21, 26, 21, 12,  5, 1, 1;
  1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1;
  ...

Ronald Cools, Encyclopaedia of Cubature Formulas
Ivan P. Mysovskikh, On the construction of cubature formulae for very simple domains, USSR Computational Mathematics and Mathematical Physics, Volume 4, Issue 1, 1964, 1-17.

Main diagonal: A000108.

Cf. A007318, A037306, A046854, A047996, A065941, A241926, A267632.

b(n, k) := (n + k)!/((n + 1)!*k!)$
T(n, k) := if integerp(b(n, k)) then b(n, k) else floor(b(n, k)) + 1$
create_list(T(k, n - k), n, 0, 15, k, 0, n);

cp's OEIS Frontend

A347974 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_8)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

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A347975 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_9)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

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A322596 Square array read by descending antidiagonals (n >= 0, k >= 0): let b(n,k) = (n+k)!/((n+1)!*k!); then T(n,k) = b(n,k) if b(n,k) is an integer, and T(n,k) = floor(b(n,k)) + 1 otherwise.

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