A215925 The number of distinct (up to unitary similarity) *-subalgebras of the n X n complex matrices.

1, 2, 5, 10, 21, 38, 72, 124, 218, 363, 607, 977, 1580, 2479, 3889, 5976, 9162, 13812, 20771, 30811, 45561, 66638, 97117, 140237, 201811, 288119, 409904, 579240, 815715, 1141916, 1593318, 2211453, 3059662, 4213395, 5784458, 7907783, 10779202, 14636771, 19819770

Offset: 0

Author

Nathaniel Johnston, Aug 27 2012

Keywords

• nonn

Comments

A *-subalgebra is a subalgebra (i.e., a subspace closed under matrix multiplication) that is closed under the conjugate transpose operation.

Also the partial sums of A006171.

Examples

			There are 10 different (up to unitary similarity) *-subalgebras of the 3 X 3 complex matrices, which contain matrices of the following forms:
0 0 0 ... a 0 0 ... a 0 0 ... a 0 0 ... a b 0
0 0 0 ... 0 0 0 ... 0 a 0 ... 0 b 0 ... c d 0
0 0 0 ... 0 0 0 ... 0 0 0 ... 0 0 0 ... 0 0 0
. . . ... . . . ... . . . ... . . . ... . . .
a 0 0 ... a 0 0 ... a 0 0 ... a b 0 ... a b c
0 a 0 ... 0 b 0 ... 0 b 0 ... c d 0 ... d e f
0 0 a ... 0 0 b ... 0 0 c ... 0 0 e ... g h i
Thus a(3) = 10.
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..1000
• William Q. Erickson, The Demazure product extended to biwords, arXiv:2407.13165 [math.CO], 2024. See p. 16.

Crossrefs

Cf. A215905, A215909, A215914.