A174586 Number of n X n (0,1) matrices with two 1's in each row having positive permanent.

0, 1, 24, 954, 59040, 5295150, 651354480, 105393619800, 21717404916480, 5554438422838200, 1726882980691176000, 641506478978753110800, 280659563041747649760000, 142843312073975729801785200, 83684308104396267184700784000, 55915646244745131440225950320000

Offset

1,3

Comments

a(n) is the normalized volume of the convex hull of (classical) parking functions of length n. - Andrés R. Vindas-Meléndez, Jan 13 2023

References

Vladimir Shevelev, On the permanent of the stochastic (0,1)-matrices with equal row sums, Izvestia Vuzov of the North-Caucasus region, Nature sciences 1 (1997), 21-38 (in Russian).

Links

Table of n, a(n) for n=1..16.

Aruzhan Amanbayeva and Danielle Wang, The convex hull of parking functions of length n, Enumer. Comb. Appl. 2 (2022), no. 2, Paper No. S2R10, 1.

Mitsuki Hanada, John Lentfer, and Andrés R. Vindas-Meléndez, Generalized parking function polytopes, arXiv:2212.06885 [math.CO], 2022.

Formula

a(2)=1, for n>=3, a(n) = A001499(n) + Sum_{k=1..n-2} (-1)^(k+1)*k!*(C(n,k))^2*(n-k)^k*a(n-k).

a(n) = n!*((n-1)/2^(n-1))*Sum_{i=0..n-2} (2i+1)!!*C(n-2,i)*(2n-1)^(n-i-2). [corrected by John Lentfer, Oct 05 2022]

For n>=2, a(n) = (n!/2^n)*Sum_{i=0..n} (2i-1)*(2i-1)!!*C(n,i)*(2n-1)^(n-i-1).

a(n) = Gamma(3/4)*(sqrt(2)*Pi*e)^(-1/2)*n!*n^(n-1/4)*(1+O(n^((-1/4)+epsilon) with arbitrary small epsilon>0 for sufficiently large n.

Mathematica

Table[n!/2^n * Sum[(2*i-1)*(2*i-1)!!*Binomial[n, i]*(2n-1)^(n-i-1), {i, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Nov 30 2017 *)

Crossrefs

Cf. A001499, A007107, A082491.

Sequence in context: A262583 A001866 A033590 * A254619 A006175 A058810

Adjacent sequences: A174583 A174584 A174585 * A174587 A174588 A174589

Keyword

nonn

Author

Vladimir Shevelev, Mar 23 2010

Status

approved