A192892 Number of n X n binary matrices whose determinants equal their permanents.

1, 2, 12, 343, 34997, 12515441, 15749457081, 72424550598849, 1282759836215548737

Offset

0,2

Comments

Lower bounded by A088672.

Similar to A145675 and A145676.

Links

Table of n, a(n) for n=0..8.

Christopher Culter, C++ code to compute large terms

Math StackExchange, What is the number of n X n binary matrices A such that det(A)=perm(A)?

Formula

a(n) <= 2^(n^2), with equality for n<=1.

Example

a(2) equals 12 because there are exactly twelve 2 X 2 binary matrices whose determinants equal their permanents; these matrices are:
|0 0|  |1 0|  |0 1|  |1 1|  |0 0|  |1 0|  |0 0|  |1 0|
|0 0|  |0 0|  |0 0|  |0 0|  |1 0|  |1 0|  |0 1|  |0 1|
.
|0 1|  |1 1|  |0 0|  |1 0|
|0 1|  |0 1|  |1 1|  |1 1|

Mathematica

Sum[KroneckerDelta[Det[Array[Mod[Floor[k/(2^(n*(#1 - 1) + #2 - 1))], 2] &, {n, n}]], Permanent[Array[Mod[Floor[k/(2^(n*(#1 - 1) + #2 - 1))], 2] &, {n, n}]]], {k, 0, (2^(n^2)) - 1}]

Prog

(Python)
from itertools import product
from sympy import Matrix
def A192892(n): return 1 if n == 0 else sum(1 for m in product([0, 1], repeat=n**2) if (lambda x:x.det()==x.per())(Matrix(n, n, m))) # Chai Wah Wu, Oct 01 2021

Crossrefs

Cf. A046747, A087983, A089472, A089477, A145675, A145676.

Sequence in context: A088229 A060596 A074257 * A105231 A009398 A009698

Adjacent sequences: A192889 A192890 A192891 * A192893 A192894 A192895

Keyword

hard,more,nonn,nice

Author

John M. Campbell, Jul 11 2011

Extensions

a(0)=1 prepended and a(5)-a(8) from Christopher Culter, Apr 13 2016

Definition and example slightly modified by Harvey P. Dale, Feb 24 2017

Status

approved