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Search: a132186 -id:a132186
Displaying 1-10 of 11 results found. page 1 2
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A086907 Duplicate of A132186. +20
0
1, 2, 8, 58, 802, 20834, 1051586, 102233986, 19614424834, 7355623374338 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..9.
CROSSREFS
Cf. A081080, A132186.
KEYWORD
dead
STATUS
approved
A290516 Number of diagonalizable n X n matrices over GF(3). +10
5
1, 3, 39, 2109, 417153, 346720179, 1233891662727, 17484682043488557, 1077565432934756756289, 290674711165255613845226787, 320439909778519092353160948081831, 1554385919734090411686737202215725913181, 33245671345010828575975932818988836416481765697 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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.
FORMULA
a(n)/A053290(n) is the coefficient of x^n in (Sum_{n>=0} x^n/A053290(n))^3.
MATHEMATICA
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]
CROSSREFS
Cf. A053290, A132186.
Row sums of A296605.
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Aug 04 2017
STATUS
approved
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. +10
5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
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.
LINKS
Table of n, a(n) for n=0..44.
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.
FORMULA
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)).
EXAMPLE
Triangle 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;
MATHEMATICA
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}]]]
CROSSREFS
Cf. A132186 (row sums).
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Dec 15 2017
STATUS
approved
A346222 Number of semi-simple n X n matrices over GF(2). +10
2
1, 2, 10, 218, 25426, 11979362, 24071588290, 195647202043778, 6352629358366433026, 829377572450912758955522, 434523953108209440907114707970, 911402584183760891982341170891585538, 7638756947617134519287879000741815013863426, 256253116935172010151547980961815772566257949204482 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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.
LINKS
Table of n, a(n) for n=0..13.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
FORMULA
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.
MATHEMATICA
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]
CROSSREFS
Cf. A001037, A002884, A132186.
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Jul 11 2021
STATUS
approved
A086922 Number of idempotent n X n (0,1) matrices over the reals. +10
1
1, 2, 8, 50, 452, 5682, 96608, 2185738, 65108492 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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)
LINKS
Table of n, a(n) for n=0..8.
CROSSREFS
Cf. A000292, A028895, A132186, A222821.
KEYWORD
nonn,more
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 19 2003
EXTENSIONS
a(5)-a(6) from Torlach Rush, Jun 17 2020
a(7)-a(8) from A222821 added by Giovanni Resta, Jun 23 2020
STATUS
approved
A342245 Number of ordered pairs (S,T) of n X n idempotent matrices over GF(2) such that ST = TS = S. +10
1
1, 3, 21, 339, 12483, 1074339, 219474243, 107174166147, 126918737362179, 367662330459585027, 2614066808849501254659, 45985259502347910886975491, 2009925824909891218929491103747, 218411680908756813835229484489351171 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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.
LINKS
Table of n, a(n) for n=0..13.
FORMULA
Let E(x) = Sum_{n>=0} x^n/(2^binomial(n,2) * [n]_2!) where [n]_2! = A005329(n). Then E(x)^3 = Sum_{n>=0} a(n)x^n/(2^binomial(n,2) * [n]_2!)
MATHEMATICA
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]
CROSSREFS
Cf. A132186, A005329.
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Mar 07 2021
STATUS
approved
A285053 Multiplications between idempotent equivalence classes for n X n matrices over GF(2). +10
0
1, 4, 118, 27080 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..4.
CROSSREFS
The idempotents are enumerated in A132186.
KEYWORD
nonn,more
AUTHOR
Chad Brewbaker, Apr 08 2017
STATUS
approved
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. +10
0
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..35.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
EXAMPLE
1;
0, 2;
2, 6, 8;
48, 196, 210, 58;
5824, 23280, 27020, 8610, 802;
2887680, 11550848, 13756560, 4757260, 581250, 20834;
MATHEMATICA
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
CROSSREFS
Cf. A002820 (column k=0), A132186 (main diagonal), A002416 (row sums).
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Jul 16 2021
STATUS
approved
A346677 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) that can be decomposed as the direct sum of k cyclic matrices, 0<=k<=n, n>=0. +10
0
1, 0, 2, 0, 8, 8, 0, 132, 322, 58, 0, 10752, 36412, 17570, 802, 0, 3185280, 16923024, 11693324, 1731970, 20834, 0, 5279662080, 26989750656, 30003846992, 6109974636, 335190786, 1051586, 0, 28343145922560, 196717668747264, 247267921788288, 84586214764240, 5906325116460, 128574848514, 102233986 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..35.
Joseph Kung, The Cycle Structure of a Linear Transformation over a Finite Field, Linear Algebra and its Applications, Vol 36, 1981, pages 141-155.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
EXAMPLE
Triangle begins:
1;
0, 2;
0, 8, 8;
0, 132, 322, 58;
0, 10752, 36412, 17570, 802;
0, 3185280, 16923024, 11693324, 1731970, 20834;
...
MATHEMATICA
nn = 6; 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_, deg_] := Total[Map[v^Length[#] u^(deg Total[#])/aut[deg, #] &, 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[Product[g[u, v, deg]^A001037[[deg]], {deg, 1, nn}], {u, 0, nn}], {u, v}])[[n]], n], {n, 1, nn}] // Grid
CROSSREFS
Cf. A002416 (row sums) A132186 (main diagonal).
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Jul 28 2021
STATUS
approved
A357410 a(n) is the number of covering relations in the poset P of n X n idempotent matrices over GF(2) ordered by A <= B if and only if AB = BA = A. +10
0
0, 1, 12, 224, 6960, 397792, 42001344, 8547291008, 3336917303040, 2565880599084544, 3852698988517260288, 11517943538435677485056, 67829192662051610706309120, 799669932659456441970547744768, 18652191511341505602408972738871296, 873360272626100960024734923878091948032 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The order given for P in the title is equivalent to the ordering: A <= B if and only if image(A) is contained in image(B) and null(A) contains null(B). Then A is covered by B if and only if there is a 1-dimensional subspace U such that image(B) = image(A) direct sum U. If such a subspace U exists then it is unique and is equal to the intersection of null(A) with image(B). The number of maximal chains in P is A002884(n). The set of all idempotent matrices over GF(q) with this ordering is a binomial poset with factorial function |GL_n(F_q)|/(q-1)^n. (see Stanley reference).
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Vol I, second edition, page 320.
LINKS
Table of n, a(n) for n=0..15.
Wikipedia, Covering relation
FORMULA
Sum_{n>=0} a(n) x^n/B(n) = x * (Sum_{n>=0} x^n/B(n))^2 where B(n) = A002884(n).
EXAMPLE
a(2) = 12 because there are A132186(2) = 8 idempotent 2 X 2 matrices over GF(2). The identity matrix covers 6 rank 1 matrices each of which covers the zero matrix for a total of 12 covering relations. Cf. A296548.
MATHEMATICA
nn = 15; B[q_, n_] := Product[q^n - q^i, {i, 0, n - 1}]/(q - 1)^n;
e[q_, u_] := Sum[u^n/B[q, n], {n, 0, nn}]; Table[B[2, n], {n, 0, nn}] CoefficientList[Series[e[2, u] u e[2, u], {u, 0, nn}], u]
CROSSREFS
Cf. A132186, A342245, A002884, A296548.
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Sep 26 2022
STATUS
approved
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Last modified August 29 21:13 EDT 2024. Contains 375518 sequences. (Running on oeis4.)