A361082 Number of 3 X 3 matrices with unit determinant and positive integer entries whose sum is n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 18, 72, 108, 234, 360, 747, 756, 1818, 1782, 3222, 3672, 6615, 5850, 11394, 11034, 16623, 17028, 30204, 22248, 45792, 39204, 56853, 57906, 87984, 72036, 128160, 108990, 154890

Offset

0,12

Comments

The analog for 2 X 2 matrices turns out to be A000010(n)-2, cf. mathoverflow post by user FFCH.

Links

Table of n, a(n) for n=0..39.

Pavel Gubkin, Number of matrices with unit determinant and fixed sum of elements, mathoverflow, Feb. 28, 2023.

User FFCH, Decomposition of a natural number as sum of positive integers, mathoverflow, Feb. 23, 2023.

Formula

a(n) / n^5 appears to have lim inf > 0.002 and lim sup < 0.004. [Observation by Brendan McKay, cf. Gubkin mathoverflow link.]

Example

a(n) = 0 for n < 11, because all 9 entries must be >= 1 and a nonzero determinant requires 3 linearly independent, thus distinct rows and columns, so we need at least two entries >= 2.
a(11) = 9 counts the 9 possible 3 X 3 matrices with all but two entries equal to 1, and the two 2's in two distinct rows and columns and such that for both of them the row index is either equal or always one less or always one more than the column index (mod 3).

Prog

(Python)
from sympy.utilities.iterables import multiset_permutations, partitions
def A361082(n):
    c = 0
    for s, d in partitions(n, m=9, k=n-8, size=True):
        if s == 9:
            c += sum(1 for p in multiset_permutations(d) if p[0]*(p[4]*p[8]-p[5]*p[7])-p[1]*(p[3]*p[8]-p[5]*p[6])+p[2]*(p[3]*p[7]-p[4]*p[6])==1)
    return c # Chai Wah Wu, Mar 02 2023

Crossrefs

Cf. A000010 (analog for 2 X 2 matrices with nonnegative entries).

Cf. A361083 (analog for 3 X 3 matrices with nonnegative entries).

Sequence in context: A232921 A295473 A255839 * A166640 A140149 A197345

Adjacent sequences: A361079 A361080 A361081 * A361083 A361084 A361085

Keyword

nonn

Author

M. F. Hasler, Mar 01 2023

Extensions

Values up to a(109) from Brendan McKay, Mar 10 2023

Status

approved