A005158 Number of alternating sign n X n matrices invariant under a half-turn.
1, 2, 3, 10, 25, 140, 588, 5544, 39204, 622908, 7422987, 198846076
Offset: 1
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
- G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001.
- 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]
Crossrefs
A059475(n) = a(2n).
Formula
Robbins gives simple (conjectured) formulas related to this sequence in Section 3.3.
a(n) = a(n-1) * (1 + [n even]/3) * C(n\2*3, n\2) / C(n\2*2, n\2) for all n > 1, where C(.,.) are the binomial coefficients, n\2 := floor(n/2) and [n even] = 1 if n is even, 0 else (Iverson bracket). [From Robbins conjectured(!) formulas.] - M. F. Hasler, Jun 15 2019