A130306 Degree of the scheme of n X n complex matrices that square to zero.
1, 2, 2, 12, 28, 440, 2456, 98448, 1327632, 134302752, 4398726432, 1116577758912, 89104889764288, 56558827752672128, 11021135122877392256, 17451895365397015875840, 8316834448188073547563264, 32799202036840274283669160448, 38271513084756431661704424923648
Offset: 0
Keywords
Examples
In size 1, the scheme {x^2=0} is of degree 2. in size 2, the scheme of matrices {{m11,m12},{m21,m22}} that square to zero is generically reduced and the corresponding reduced scheme is given by the equations m11+m22=0 and m11^2+m12 m21=0, hence also of degree 2.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..100
- A. Knutson and P. Zinn-Justin, A scheme related to the Brauer loop model, Adv. in Math. 214 (2007), 40-77.
Crossrefs
Cf. A130294.
Programs
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Mathematica
a[1] = 2; a[n_] := If[EvenQ[n], 2^(n/2) Det[Table[Binomial[2i + 2j + 1, 2i], {i, 0, n/2-1}, {j, 0, n/2-1}]], 2^((n-1)/2+1) Det[Table[Binomial[2i + 2j + 3, 2i + 1], {i, 0, (n-1)/2-1}, {j, 0, (n-1)/2-1}]]]; Array[a, 12] (* Jean-François Alcover, Dec 04 2018 *)
Formula
a(2n) = 2^n * det(binomial(2i+2j+1,2i)), 0<=i,j<=n-1; a(2n+1) = 2^(n+1) * det(binomial(2i+2j+3,2i+1)), 0<=i,j<=n-1.
Extensions
More terms from Alois P. Heinz, Dec 04 2018