A005163 Number of alternating sign n X n matrices that are symmetric about a diagonal.

1, 2, 5, 16, 67, 368, 2630, 24376, 293770, 4610624, 94080653, 2492747656, 85827875506, 3842929319936, 223624506056156, 16901839470598576, 1659776507866213636, 211853506422044996288, 35137231473111223912310, 7569998079873075147860464

Offset: 1

Author

N. J. A. Sloane and Simon Plouffe

Keywords

• nonn
• easy
• nice

Comments

Robbins's paper does not give a formula for this sequence. On the contrary he states: "Apparently these numbers do not factor into small primes, so a simple product formula seems unlikely. Of course this does not rule out other very simple formulas, but these would be more difficult to discover (let alone prove)." As far as I know no formula is currently known. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.

Links

• Christoph Koutschan, Table of n, a(n) for n = 1..131
• Roger E. Behrend, Ilse Fischer, and Christoph Koutschan, Diagonally symmetric alternating sign matrices, arXiv:2309.08446 [math.CO], 2023.
• Mireille Bousquet-Mélou and Laurent Habsieger, Sur les matrices à signes alternants, [On alternating-sign matrices] in Formal power series and algebraic combinatorics (Montreal, PQ, 1992). Discrete Math. 139 (1995), 57-72.
• D. P. Robbins, Symmetry classes of alternating sign matrices, arXiv:math/0008045 [math.CO], 2000.
• R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]

Extensions

More terms (taken from Bousquet-Mélou & Habsieger's paper) from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008