A192892 Number of n X n binary matrices whose determinants equal their permanents.
1, 2, 12, 343, 34997, 12515441, 15749457081, 72424550598849, 1282759836215548737
Offset: 0
Examples
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|
Links
- 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)?
Programs
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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}] -
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
Formula
a(n) <= 2^(n^2), with equality for n<=1.
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
Comments