A132186 -id:A132186 - cp's OEIS frontend

A086907 Duplicate of A132186.

1, 2, 8, 58, 802, 20834, 1051586, 102233986, 19614424834, 7355623374338

Offset: 0

dead

A296548 Triangle read by rows: T(n,k) is the number of diagonalizable n X n matrices over GF(2) that have rank k, n >= 0, 0 <= k <= n.

1, 1, 1, 1, 6, 1, 1, 28, 28, 1, 1, 120, 560, 120, 1, 1, 496, 9920, 9920, 496, 1, 1, 2016, 166656, 714240, 166656, 2016, 1, 1, 8128, 2731008, 48377856, 48377856, 2731008, 8128, 1, 1, 32640, 44216320, 3183575040, 13158776832, 3183575040, 44216320, 32640, 1

Offset: 0

Geoffrey Critzer, Dec 15 2017

Equivalently, T(n,k) is the number of n X n matrices, P, over GF(2) with rank k, such that P^2 = P.

Equivalently, T(n,k) is the number of direct sum decompositions of the vector space GF(2)^n into exactly two subspaces U and W such that the dimension of U is k.

			Triangle T(n,k) begins:
  1;
  1,    1;
  1,    6,      1;
  1,   28,     28,      1;
  1,  120,    560,    120,      1;
  1,  496,   9920,   9920,    496,    1;
  1, 2016, 166656, 714240, 166656, 2016, 1;
  ...

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A132186 (row sums).

b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, b(n-1, k-1)+2^k*b(n-1, k)))
    end:
T:= (n,k)-> 2^(k*(n-k))*b(n, k):
seq(seq(T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Dec 02 2024

nn = 8; g[n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[
    QFactorial[n, q]] /. q -> 2; Grid[Map[Select[#, # > 0 &] &,
  Table[g[n], {n, 0, nn}] CoefficientList[Series[Sum[(u z)^r/g[r] , {r, 0, nn}] Sum[z^r/g[r], {r, 0, nn}], {z, 0, nn}], {z, u}]]]

T(n,k)/A002884(n) is the coefficient of y^k*x^n in the expansion of Sum_{n>=0} x^n\A002884(n) * Sum_{n>=0} y*x^n\A002884(n).

T(n,k) = A002884(n)/(A002884(k)*A002884(n-k)) = A022166(n,k)*2^(k(n-k)).

A290516 Number of diagonalizable n X n matrices over GF(3).

Original entry on oeis.org

1, 3, 39, 2109, 417153, 346720179, 1233891662727, 17484682043488557, 1077565432934756756289, 290674711165255613845226787, 320439909778519092353160948081831, 1554385919734090411686737202215725913181, 33245671345010828575975932818988836416481765697

Offset: 0

Geoffrey Critzer, Aug 04 2017

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..56
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A053290, A132186.

Row sums of A296605.

nn = 12; g[ n_] := (q - 1)^n q^Binomial[n, 2] FunctionExpand[
    QFactorial[n, q]] /. q -> 3; G[z_] := Sum[z^k/g[k], {k, 0, nn}];Table[g[n], {n, 0, nn}] CoefficientList[Series[G[z]^3, {z, 0, nn}], z]

a(n)/A053290(n) is the coefficient of x^n in (Sum_{n>=0} x^n/A053290(n))^3.

A342245 Number of ordered pairs (S,T) of n X n idempotent matrices over GF(2) such that ST = TS = S.

Original entry on oeis.org

1, 3, 21, 339, 12483, 1074339, 219474243, 107174166147, 126918737362179, 367662330459585027, 2614066808849501254659, 45985259502347910886975491, 2009925824909891218929491103747, 218411680908756813835229484489351171, 59296916710446845619466630380450779971587

Offset: 0

Geoffrey Critzer, Mar 07 2021

nonn

The components in the ordered pairs are not necessarily distinct.

The relation S<=T iff ST=TS=S gives a partial ordering on the idempotent matrices enumerated in A132186. Each length k chain (from bottom to top) in the poset corresponds to an ordered direct sum decomposition of GF(2)^n into exactly k subspaces.

Cf. A132186, A005329.

nn = 13; b[n_] := q^Binomial[n, 2] FunctionExpand[QFactorial[n, q]] /. q -> 2;
e[x_] := Sum[x^n/b[n], {n, 0,nn}];Table[b[n],{n,0,nn}]CoefficientList[Series[e[x]^3, {x, 0, nn}], x]

Let E(x) = Sum_{n>=0} x^n/(2^binomial(n,2) * [n]A005329(n).%20Then%20E(x)%5E3%20=%20Sum">2!) where [n]_2! = A005329(n). Then E(x)^3 = Sum{n>=0} a(n)x^n/(2^binomial(n,2) * [n]_2!)

A346222 Number of semi-simple n X n matrices over GF(2).

Original entry on oeis.org

1, 2, 10, 218, 25426, 11979362, 24071588290, 195647202043778, 6352629358366433026, 829377572450912758955522, 434523953108209440907114707970, 911402584183760891982341170891585538, 7638756947617134519287879000741815013863426, 256253116935172010151547980961815772566257949204482

Offset: 0

Geoffrey Critzer, Jul 11 2021

nonn

Equivalently, number of n X n matrices over GF(2) that are diagonalizable over the algebraic closure of GF(2).

Equivalently, the number of n X n matrices over GF(2) whose minimal polynomial is a product of distinct irreducible factors, i.e., the minimal polynomial is squarefree.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A001037, A002884, A132186.

nn = 13; q = 2; A001037 =Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; \[Gamma]q[j_, d_] :=Table[Product[(q^d)^n - (q^d)^i, {i, 0, n - 1}], {n, 1, nn}][[j]];Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[ Series[Product[(1 + Sum[u^(j d)/\[Gamma]q[j, d], {j, 1, nn}])^
     A001037[[d]], {d, 1, nn}], {u, 0, nn}], u]

Sum_{n>=0} a(n)x^n/A002884(n) = Product_{d>=1} Sum_{j>=1} x^(j*d)/|GL_j(F_2^d)|)^A001037(d) where |GL_j(F_2^d)| is the order of the general linear group of degree j over the field with 2^d elements.

A381899 Irregular triangular array read by rows. T(n,k) is the number of length n words x on {0,1} such that I(x) + W(x)*(n-W(x)) = k, where I(x) is the number of inversions in x and W(x) is the number of 1's in x, n >= 0, 0 <= k <= floor(n^2/2).

Original entry on oeis.org

1, 2, 2, 1, 1, 2, 0, 2, 2, 2, 2, 0, 0, 2, 3, 3, 4, 1, 1, 2, 0, 0, 0, 2, 2, 4, 4, 6, 4, 4, 2, 2, 2, 0, 0, 0, 0, 2, 2, 2, 4, 5, 7, 6, 9, 7, 7, 5, 4, 1, 1, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 4, 4, 8, 6, 10, 12, 14, 12, 14, 10, 10, 6, 4, 2, 2

Offset: 0

Geoffrey Critzer, Mar 09 2025

Sum_{k>=0} T(n,k)*2^k = A132186(n).
Sum_{k>=0} T(n,k)*3^k = A053846(n).
Sum_{k>=0} T(n,k)*q^k = the number of idempotent n X n matrices over GF(q).
It appears that if n is even the n-th row converges to 2,0,0,...,21,13,9,5,4,1,1 which is A226622 reversed, and if n is odd the sequence is twice A226635.
From Alois P. Heinz, Mar 09 2025: (Start)
Sum_{k>=0} k * T(n,k) = 3*A001788(n-1) for n>=1.
Sum_{k>=0} (-1)^k * T(n,k) = A060546(n). (End)

			Triangle T(n,k) begins:
  1;
  2;
  2, 1, 1;
  2, 0, 2, 2, 2;
  2, 0, 0, 2, 3, 3, 4, 1, 1;
  2, 0, 0, 0, 2, 2, 4, 4, 6, 4, 4, 2, 2;
  ...
T(4,5) = 3 because we have: {0, 1, 0, 0}, {0, 1, 0, 1}, {1, 1, 0, 1}.

Alois P. Heinz, Rows n = 0..50, flattened

Row sums give A000079.

Cf. A001788, A060546, A226635, A226622, A132186, A053846, A083906.

b:= proc(i, j) option remember; expand(`if`(i+j=0, 1,
     `if`(i=0, 0, b(i-1, j))+`if`(j=0, 0, b(i, j-1)*z^i)))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(
         expand(add(b(n-j, j)*z^(j*(n-j)), j=0..n))):
seq(T(n), n=0..10);  # Alois P. Heinz, Mar 09 2025

nn = 7; B[n_] := FunctionExpand[QFactorial[n, q]]*q^Binomial[n, 2];e[z_] := Sum[z^n/B[n], {n, 0, nn}]; Map[CoefficientList[#, q] &, Table[B[n], {n, 0, nn}] CoefficientList[Series[e[z]^2, {z, 0, nn}],z]]

Sum_{n>=0} Sum_{k>=0} T(n,k)*q^k*x^n/(n_q!*q^binomial(n,2)) = e(x)^2 where e(x) = Sum_{n>=0} x^n/(n_q!*q^binomial(n,2)) where n_q! = Product{i=1..n} (q^n-1)/(q-1).

A086922 Number of idempotent n X n (0,1) matrices over the reals.

Original entry on oeis.org

1, 2, 8, 50, 452, 5682, 96608, 2185738, 65108492

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 19 2003

From Torlach Rush, Jun 18 2020: (Start)
Let m(n,k) be the number of idempotent n X n (0,1) matrices with k entries equal to 1. Then:
k    | m(n,k)
-----|------------------------------------------------------
0    | 1
1    | n
2    | A028895(n - 1)
3    | 19 * A000292(n - 2)
4    | ((n - 3) (n - 2) (n - 1) (35 n - 124))/8
5    | ((n - 4) (n - 3) (n - 2) (n - 1) (631 n - 2675))/120
...
Conjecture: There is no closed form expression for this sequence.
(End)

Cf. A000292, A028895, A132186, A222821.

a(5)-a(6) from Torlach Rush, Jun 17 2020

a(7)-a(8) from A222821 added by Giovanni Resta, Jun 23 2020

A346201 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) such that the sum of the dimensions of their eigenspaces taken over all eigenvalues is k, 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 0, 2, 2, 6, 8, 48, 196, 210, 58, 5824, 23280, 27020, 8610, 802, 2887680, 11550848, 13756560, 4757260, 581250, 20834, 5821595648, 23286380544, 28097284992, 10075582800, 1369706604, 67874562, 1051586, 47317927329792, 189271709384704, 229853403924480, 83865929653632, 11957394226896, 668707460652, 14779207170, 102233986

Offset: 0

Geoffrey Critzer, Jul 16 2021

			        1;
        0,        2;
        2,        6,        8;
       48,      196,      210,      58;
     5824,    23280,    27020,    8610,    802;
  2887680, 11550848, 13756560, 4757260, 581250, 20834;

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A002820 (column k=0), A132186 (main diagonal), A002416 (row sums).

nn = 8; q = 2; b[p_, i_] := Count[p, i];d[p_, i_] :=  Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}];aut[deg_, p_] :=  Product[Product[q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; A001037 =
Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}];g[u_, v_] :=
Total[Map[v^Length[#] u^Total[#]/aut[1, #] &,Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]]];Table[Take[(Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[g[u, v] g[u, v] Product[Product[1/(1 - (u/q^r)^d), {r, 1, \[Infinity]}]^A001037[[d]], {d, 2, nn}], {u, 0, nn}], {u, v}])[[n]],
   n], {n, 1, nn}] // Grid

A358649 Number of convergent n X n matrices over GF(2).

Original entry on oeis.org

1, 2, 11, 205, 14137, 3755249, 3916674017, 16190352314305, 266479066904477569, 17503939768635307654913, 4593798697440979773283368449, 4819699338906053452395454422580225, 20221058158328101246044232181365184919553

Offset: 0

Geoffrey Critzer, Nov 26 2022

nonn

A matrix A over a finite field is convergent if A^j=A^(j+1) for some j>=1. Every convergent matrix converges to an idempotent matrix. Every idempotent matrix is convergent to itself. Every nilpotent matrix is convergent to the zero matrix.

Cf. A296548, A053763, A132186, A379778.

nn = 12; q = 2;g[n_] := Product[q^n - q^i, {i, 0, n - 1}]; Table[Sum[g[n]/(g[k] g[n - k]) q^((n - k) (n - k - 1)), {k, 0, n}], {n, 0, nn}]

a(n) = Sum_{k=0..n} A296548(n,k)*A053763(n-k).

Sum_{n>=0} a(n)*x^n/B(n) = f(x)*e(x) where f(x)=Sum_{n>=0} q^(n^2-n)*x^n/B(n), e(x)=Sum_{n>=0} x^n/B(n), B(n)=Product_{i=0..n-1} (q^n-q^i)/(q-1)^n, and q=2. - Geoffrey Critzer, Jan 02 2025

A285053 Multiplications between idempotent equivalence classes for n X n matrices over GF(2).

Original entry on oeis.org

1, 4, 118, 27080

Offset: 1

Chad Brewbaker, Apr 08 2017

With the n X n matrices over GF(2) construct 3-tuples (a,b,c) where a*b = c. Map each of the three elements to their idempotent under self multiplication. Filter on unique 3-tuples.

The idempotents are enumerated in A132186.

cp's OEIS Frontend

A086907 Duplicate of A132186.

Views

Author

Keywords

Crossrefs

A296548 Triangle read by rows: T(n,k) is the number of diagonalizable n X n matrices over GF(2) that have rank k, n >= 0, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A290516 Number of diagonalizable n X n matrices over GF(3).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A342245 Number of ordered pairs (S,T) of n X n idempotent matrices over GF(2) such that ST = TS = S.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A346222 Number of semi-simple n X n matrices over GF(2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A381899 Irregular triangular array read by rows. T(n,k) is the number of length n words x on {0,1} such that I(x) + W(x)*(n-W(x)) = k, where I(x) is the number of inversions in x and W(x) is the number of 1's in x, n >= 0, 0 <= k <= floor(n^2/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A086922 Number of idempotent n X n (0,1) matrices over the reals.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A346201 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) such that the sum of the dimensions of their eigenspaces taken over all eigenvalues is k, 0 <= k <= n, n >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A358649 Number of convergent n X n matrices over GF(2).

Views

Author

Keywords

Comments

Crossrefs