Frank Garvan’s Counterexample to the Folsom-Ono Andrews SPT Parity Theorem

Can Be Discovered in Less Than 17 seconds Shalosh B. EKHAD ^{1}

Frank Garvan informed my master, Doron Zeilberger, that the statement of Theorem 1.2 in [FO] is false as stated. A corrected version will appear in [AGL]. This got me curious how long would it take me to discover this error, had I been asked to find it. My master spent a few minutes writing a Maple code available from

http://www.math.rutgers.edu/~zeilberg/tokhniot/CheckAmandaKen .

Once copied into a Maple session, typing

CheckKenAmanda(507);

returns (in 16.28 seconds):

false .

(Note that 24 · 507 − 1 = 23 · (23)^{2}, but spt(507) is even). Furthermore, if the statement is already false, how can the proof be correct? Alas, I have better things to do than try and fix their proof to fit the corrected statement in [AGL], and besides what is the point? [AGL] give a brand new proof that is much more appealing (at least to me) since it is combinatorial and elementary, and does not use weak Maass forms and other fancy stuff. References

[AGL] G.E. Andrews, F. G. Garvan, and J. Liang, Self-conjugate vector partitions and the parity of the spt-function, in preparation.

[FO] A. Folsom and K. Ono, The spt-function of Andrews, Proc. Natl. Acad. Sci. USA 105 (2008), 20152-20156.

1 c/o zeilberg at math dot rutgers dot edu.

Exclusively published in: http://www.math.rutgers.edu/~zeilberg/pj.html . Oct. 28, 2011.