A125587 -id:A125587 - cp's OEIS frontend

A125593 Number of nonsingular real n X n {0,1}-matrices which are not robust (cf. A125587).

Original entry on oeis.org

0, 2, 106, 17552, 10911088, 26612379360

Offset: 1

Artur Jasinski, Jan 07 2007

Cf. A055165, A125587, A126603.

a(n) = A055165(n) - A125587(n).

Typo in a(5) corrected by Giovanni Resta, Jul 23 2025

A126603 Number of real {0,1}-matrices of size n X n which are not robust (cf. A125587).

Original entry on oeis.org

1, 12, 444, 60528, 31951200, 66495244096

Offset: 1

Artur Jasinski, Jan 06 2007

a(n) = 2^(n^2) - A125587(n) = A002416(n) - A125587(n). - Klaus Brockhaus, Jan 08 2007

Cf. A125587, A002416.

a(5) and a(6) (from A125587) from Klaus Brockhaus, Jan 08 2007

A127706 Number of nonsingular n X n real {0,1}-matrices n X n which are not robust (cf. A125587) nor in A127186.

Original entry on oeis.org

0, 0, 66, 13616, 9215792

Offset: 1

Artur Jasinski, Jan 24 2007

Cf. A055165, A125587, A127186, A125593.

a(n) = A055165(n) - A125587(n) - A127186(n), for n > 1.

Name and a(4) corrected by Giovanni Resta, Jul 23 2025

A127182 Number of distinct characteristic polynomials of n X n real robust {0,1}-matrices.

Original entry on oeis.org

1, 2, 12, 156, 5612

Offset: 1

Artur Jasinski and Peter Pein (petsie(AT)dordos.net), Jan 07 2007

nonn

			a(2)=2 because there are 4 binary robust 2 X 2 matrices, but only two distinct characteristic polynomials, namely y^2-y-1 and y^2-2y+1.
a(3)=12 because there are 12 different characteristic polynomials: -1-3y-y^2+y^3, 1-2y-y^2+y^3, 1+y-3y^2+y^3, -2+3y-3y^2+y^3, -1-2y-y^2+y^3, -1-y-y^2+y^3, -1+y-2y^2+y^3, 2-y-2y^2+y^3, -1+2y-3y^2+y^3, 1-y-2y^2+y^3, 1- 2y^2+y^3, -1+3y-3y^2+y^3.

Cf. A125587, A125593, A127183, A127184, A127186.

mats[1] = {{{1}}}; mats[n_Integer?Positive] := mats[n] = Module[{newrows = Rest[Tuples[{0, 1}, {n}]], mp1 = Flatten[Function[k, Thread[(Append[ #1, #2]&)[ #1, k]]& /@ mats[n - 1]] /@ Tuples[{0, 1}, {n - 1}], 1]}, Flatten[MapThread[Function[{m, nl}, Append[m, # ]& /@ nl], {mp1, Pick[newrows, # =!= 0& /@ # ]& /@ (First /@ Dot[NullSpace /@ mp1, Transpose[newrows]])}], 1]] A127182[n_]=Length[Union[CharacteristicPolynomial[mats[n]]]]

A127186 Number of nonsingular n X n real {0,1}-matrices such that all upper left i X i submatrices are singular for i = 1..n-1.

Original entry on oeis.org

1, 2, 40, 3936, 1695296

Offset: 1

Artur Jasinski, Jan 08 2007

Cf. A125587, A125593, A127182, A127183.

check[x_,n_] := Catch@ Block[{m = Partition[ IntegerDigits[x, 2, n^2], n]}, Do[ If[ Det[m[[1 ;; k, 1 ;; k]]] != 0, Throw@ False], {k, n-1}]; Det[m] != 0]; a[n_] := Length@ Parallelize@ Select[ Range[0, 2^(n^2) - 1], check[#, n] &]; Array[a, 4] (* Giovanni Resta, Jul 23 2025 *)

a(1) and a(4) corrected by Giovanni Resta, Jul 23 2025

A127183 Number of distinct irreducible characteristic polynomials of n X n real robust {0,1}-matrices.

Original entry on oeis.org

1, 1, 6, 88, 3719

Offset: 1

Artur Jasinski and Peter Pein (petsie(AT)dordos.net), Jan 07 2007

nonn

			a(3)=6 because there are 6 different irreducible characteristic polynomials of 3 x 3 binary robust matrices: -1-2y-y^2+y^3,-1-2y-y^2+y^3, -1-y-y^2+y^3, -1+y-2y^2+y^3, -1+2y-3y^2+y^3, 1-y-2y^2+y^3.

Cf. A125587, A125593, A127182, A127184, A127186.

a(n) = A127182(n) - A127184(n).

A125586 a(n) = 2^(2n-1) - (n+2)*3^(n-2).

Original entry on oeis.org

1, 4, 17, 74, 323, 1400, 6005, 25478, 107015, 445556, 1841273, 7561922, 30897227, 125714672, 509767421, 2061390206, 8317305359, 33498803948, 134727010049, 541232563130, 2172291241811, 8712410196584, 34922863258757, 139921580805494, 560408087592983

Offset: 1

N. J. A. Sloane and Vinay Vaishampayan, Jan 05 2007

Number of n X n nonsingular real matrices with entries {0,1} in which the top left n-1 X n-1 submatrix is the identity matrix. See A125587 for proof.

The number of singular matrices is given by A006234.

			a(2) = 4:
10 10 11 11
01 11 01 10

Index entries for linear recurrences with constant coefficients, signature (10,-33,36).

Cf. A125587, A006234.

A125586:=n->2^(2n-1)-(n+2)*3^(n-2); seq(A125586(n), n=1..30); # Wesley Ivan Hurt, Feb 26 2014

Table[2^(2n-1)-(n+2)*3^(n-2), {n, 30}] (* Wesley Ivan Hurt, Feb 26 2014 *)
LinearRecurrence[{10,-33,36},{1,4,17},50] (* Harvey P. Dale, Sep 15 2019 *)

Vec(-x*(10*x^2-6*x+1)/((3*x-1)^2*(4*x-1)) + O(x^100)) \\ Colin Barker, Feb 26 2014

G.f.: -x*(10*x^2-6*x+1) / ((3*x-1)^2*(4*x-1)). - Colin Barker, Feb 26 2014

A127184 Number of distinct reducible characteristic polynomials of n X n real robust {0,1}-matrices.

Original entry on oeis.org

0, 1, 6, 68, 1893

Offset: 1

Artur Jasinski and Peter Pein (petsie(AT)dordos.net), Jan 07 2007

nonn

Cf. A125587, A125593, A127182, A127183, A127186.

a(n)=A127182(n)-A127183(n)

cp's OEIS Frontend

A125593 Number of nonsingular real n X n {0,1}-matrices which are not robust (cf. A125587).

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A126603 Number of real {0,1}-matrices of size n X n which are not robust (cf. A125587).

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A127706 Number of nonsingular n X n real {0,1}-matrices n X n which are not robust (cf. A125587) nor in A127186.

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A127182 Number of distinct characteristic polynomials of n X n real robust {0,1}-matrices.

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A127186 Number of nonsingular n X n real {0,1}-matrices such that all upper left i X i submatrices are singular for i = 1..n-1.

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A127183 Number of distinct irreducible characteristic polynomials of n X n real robust {0,1}-matrices.

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A125586 a(n) = 2^(2n-1) - (n+2)*3^(n-2).

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A127184 Number of distinct reducible characteristic polynomials of n X n real robust {0,1}-matrices.

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