A362679 a(n) is the permanent of the n X n symmetric matrix M(n) defined by M[i, j, n] = min(i, j)*(n + 1) - i*j.

1, 1, 5, 72, 2309, 140400, 14495641, 2347782144, 562385930985, 190398813728000, 87889475202276461, 53726132414026874880, 42454821207656237294381, 42495322215073539046387712, 52954624815227996007075890625, 80932107560443542398970529579008, 149736953621087625813286348913927569

Offset: 0

Author

Stefano Spezia, Apr 29 2023

Keywords

• nonn

Comments

M(n-1)/n is the inverse of the Cartan matrix for SU(n): the special unitary group of degree n.

The elements sum of the matrix M(n) is A002415(n+1).

The antidiagonal sum of the matrix M(n) is A005993(n-1).

The n-th row of A107985 gives the row or column sums of the matrix M(n+1).

Examples

			a(3) = 72:
           [3, 2, 1]
    M(3) = [2, 4, 2]
           [1, 2, 3]
a(5) = 140400:
           [5, 4, 3, 2, 1]
           [4, 8, 6, 4, 2]
    M(5) = [3, 6, 9, 6, 3]
           [2, 4, 6, 8, 4]
           [1, 2, 3, 4, 5]
		

References

• E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Am. Math. Soc. Translations, Series 2, Vol. 6, 1957.

Links

• Chao Ju, Chern-Simons Theory, Ehrhart Polynomials, and Representation Theory, arXiv:2304.11830 [math-ph], 2023. See p. 14.
• Stefano Spezia, A determinantal formula for the number of trees on n labeled nodes
• Wikipedia, Special unitary group

Crossrefs

Cf. A000272, A000292 (trace), A002415, A003983, A003991, A005993, A106314 (antidiagonals), A107985.

Programs

Maple

a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
    Matrix(n, (i, j)-> min(i, j)*(n+1)-i*j))):
seq(a(n), n=0..16);  # Alois P. Heinz, Apr 30 2023

Mathematica

M[i_, j_, n_]:=Min[i, j](n+1)-i j; Join[{1}, Table[Permanent[Table[M[i, j, n], {i, n}, {j, n}]], {n, 17}]]

PARI

a(n) = matpermanent(matrix(n, n, i, j, min(i, j)*(n + 1) - i*j)); \\ Michel Marcus, Apr 30 2023

Formula

Conjecture: det(M(n)) = A000272(n+1).

The conjecture is true (see proof in Links). - Stefano Spezia, May 24 2023