A346214 -id:A346214 - cp's OEIS frontend

A053763 a(n) = 2^(n^2 - n).

1, 1, 4, 64, 4096, 1048576, 1073741824, 4398046511104, 72057594037927936, 4722366482869645213696, 1237940039285380274899124224, 1298074214633706907132624082305024, 5444517870735015415413993718908291383296, 91343852333181432387730302044767688728495783936

Offset: 0

Stephen G Penrice, Mar 29 2000

Nilpotent n X n matrices over GF(2). Also number of simple digraphs (without self-loops) on n labeled nodes (see also A002416).
For n >= 1 a(n) is the size of the Sylow 2-subgroup of the Chevalley group A_n(4) (sequence A053291). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001
(-1)^ceiling(n/2) * resultant of the Chebyshev polynomial of first kind of degree n and Chebyshev polynomial of first kind of degree (n+1) (cf. A039991). - Benoit Cloitre, Jan 26 2003
The number of reflexive binary relations on an n-element set. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005
From Rick L. Shepherd, Dec 24 2008: (Start)
Number of gift exchange scenarios where, for each person k of n people,
i) k gives gifts to g(k) of the others, where 0 <= g(k) <= n-1,
ii) k gives no more than one gift to any specific person,
iii) k gives no single gift to two or more people and
iv) there is no other person j such that j and k jointly give a single gift.
(In other words -- but less precisely -- each person k either gives no gifts or gives exactly one gift per person to 1 <= g(k) <= n-1 others.) (End)
In general, sequences of the form m^((n^2 - n)/2) enumerate the graphs with n labeled nodes with m types of edge. a(n) therefore is the number of labeled graphs with n nodes with 4 types of edge. To clarify the comment from Benoit Cloitre, dated Jan 26 2003, in this context: simple digraphs (without self-loops) have four types of edge. These types of edges are as follows: the absent edge, the directed edge from A -> B, the directed edge from B -> A and the bidirectional edge, A <-> B. - Mark Stander, Apr 11 2019

			a(2)=4 because there are four 2 x 2 nilpotent matrices over GF(2):{{0,0},{0,0}},{{0,1},{0,0}},{{0,0},{1,0}},{{1,1,},{1,1}} where 1+1=0. - _Geoffrey Critzer_, Oct 05 2012

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 521.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 5, Eq. (1.1.5).

T. D. Noe, Table of n, a(n) for n = 0..35
Marcus Brinkmann, Extended Affine and CCZ Equivalence up to Dimension 4, Ruhr University Bochum (2019).
N. J. Fine and I. N. Herstein, The probability that a matrix be nilpotent, Illinois J. Math., 2 (1958), 499-504.
Murray Gerstenhaber, On the number of nilpotent matrices with coefficients in a finite field, Illinois J. Math., Vol. 5 (1961), 330-333.
Antal Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
Pakawut Jiradilok, Some Combinatorial Formulas Related to Diagonal Ramsey Numbers, arXiv:2404.02714 [math.CO], 2024. See p. 19.
Greg Kuperberg, Symmetry classes of alternating-sign matrices under one roof, Annals of mathematics, Second Series, Vol. 156, No. 3 (2002), pp. 835-866, arXiv preprint, arXiv:math/0008184 [math.CO], 2000-2001 (see Th. 3).
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Götz Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

Row sums of A123554, A189898, A346412, A346214.

Cf. A053773, A006125, A000273, A000984, A002416, A319016.

seq(4^(binomial(n, n-2)), n=0..12); # Zerinvary Lajos, Jun 16 2007
a:=n->mul(4^j, j=1..n-1): seq(a(n), n=0..12); # Zerinvary Lajos, Oct 03 2007

Table[2^(2*Binomial[n,2]), {n,0,20}] (* Geoffrey Critzer, Oct 04 2012 *)

a(n)=1<<(n^2-n) \\ Charles R Greathouse IV, Nov 20 2012

def A053763(n): return 1<Chai Wah Wu, Jul 05 2024

Sequence given by the Hankel transform (see A001906 for definition) of A059231 = {1, 1, 5, 29, 185, 1257, 8925, 65445, 491825, ...}; example: det([1, 1, 5, 29; 1, 5, 29, 185; 5, 29, 185, 1257; 29, 185, 1257, 8925]) = 4^6 = 4096. - Philippe Deléham, Aug 20 2005
a(n) = 4^binomial(n, n-2). - Zerinvary Lajos, Jun 16 2007
a(n) = Sum_{i=0..n^2-n} binomial(n^2-n, i). - Rick L. Shepherd, Dec 24 2008
G.f. A(x) satisfies: A(x) = 1 + x * A(4*x). - Ilya Gutkovskiy, Jun 04 2020
Sum_{n>=1} 1/a(n) = A319016. - Amiram Eldar, Oct 27 2020
Sum_{n>=0} a(n)*u^n/A002884(n) = Product_{r>=1} 1/(1-u/q^r). - Geoffrey Critzer, Oct 28 2021

A358433 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) with index k, n>=1, 1<=k<=n.

Original entry on oeis.org

2, 13, 3, 365, 105, 42, 43801, 12915, 6300, 2520, 21725297, 6412815, 3228960, 1562400, 624960, 43798198753, 12928608063, 6533019360, 3254791680, 1574899200, 629959680, 355991759464385, 105083758588095, 53109556520832, 26576858972160, 13227473387520, 6400390348800, 2560156139520

Offset: 1

Geoffrey Critzer, Nov 15 2022

The index of a matrix A is the smallest positive integer such that rank(A^k) = rank(A^(k+1)).

			      2,
      13,       3,
     365,     105,      42,
   43801,   12915,    6300,    2520,
21725297, 6412815, 3228960, 1562400, 624960,

Cf. A002416 (row sums), A348015 (column k=1), A083402 (main diagonal for n>1), A346214.

nn = 6; q = 2; b[p_, i_] := Count[p, i];s[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^(s[p, i] deg) - q^((s[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; \[Nu] = Table[1/n Sum[MoebiusMu[n/m] q^m, {m, Divisors[n]}], {n, 1, nn}];
l[greatestpart_] :=Level[Table[IntegerPartitions[n, {0, n}, Range[greatestpart]], {n, 0,nn}], {2}];g1[u_, v_, deg_] := Total[Map[v ^(If[ Max[Prepend[#, 0]] == 0, 1, Max[Prepend[#, 0]]]) u^(deg Total[#])/aut[deg, #] &, l[nn]]]; Map[Select[#, # > 0 &] &,Drop[Table[Product[q^n - q^i, {i, 0, n - 1}], {n,0,nn}]CoefficientList[
  Series[g1[u, v, 1] g1[u, 1, 1]^(q - 1) Product[g1[u, 1, d]^\[Nu][[d]], {d, 2, nn}], {u, 0, nn}], {u, v}], 1]] // Grid

A384038 Number of 2n X 2n matrices M over GF(2) such that the column space of M is equal to the null space of M.

Original entry on oeis.org

1, 3, 210, 234360, 4047865920, 1092146608143360, 4650098142288472473600, 314462403262051153026062745600, 338960040818652280796119613717033779200, 5834618256563872511581456247120956565738854809600, 1605370810586153268821245248112723240374305354675084328960000

Offset: 0

Geoffrey Critzer, May 17 2025

nonn

Let M be a 2n X 2n matrix over GF(2) such that the column space of M is equal to the null space of M. Then M is idempotent and nullity(M) = n and index(M) = 2. If M' is similar to M then the column space of M' equals the null space of M'. Moreover, all such matrices are in the same similarity class (see Hoffman link).

			a(1) = 3 because there are 3 matrices of size 2 X 2 over GF(2) with the desired property: {{0, 0}, {1, 0}}, {{0, 1}, {0, 0}}, {{1, 1}, {1, 1}}.

D. G. Hoffman, Digraphs of finite linear transformations, Australasion Journal of Combinatorics,12: 225-238 (1995).

Cf. A002884, A006098, A346214, A053763.

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]}]; Table[Product[2^(2 k) - 2^i, {i, 0, (2 k) - 1}]/aut[1, Table[2, {k}]], {k,0, 10}]

a(n) = A002884(n)*A006098(n).

cp's OEIS Frontend

A053763 a(n) = 2^(n^2 - n).

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A358433 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) with index k, n>=1, 1<=k<=n.

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Mathematica

A384038 Number of 2n X 2n matrices M over GF(2) such that the column space of M is equal to the null space of M.

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Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula