Definition:

The shuffle algebra is a sub-algebra of the symmetric group algebra. It is simply the image of the descent algebra under the anti-isomorphism $ \theta:\mathbb{Q}[S_n]\rightarrow\mathbb{Q}[S_n]$ which sends a permutation, in $S_n$, to its inverse.

Références:

• H.N. Minh, M. Petitot, and J. Van Der Hoven, Shuffle algebra and polylogarithms, Discrete Math, Volume 225, Issues 1–3, 28, 2000, 217–230.
• D. Bayer and P. Diaconis, Trailing the Dovetail Shuffle to its Lair, The annals of Applied Probability, 1992, Vol.2, No.2, 294-313.
• G. Mélançon and C. Reutenauer, Lyndon words, Free Algebras and Shuffles, Canadian Journal of Mathematics, Vol. XLI, No. 4, 1989, 577-591.
• P. Diaconis, R.L. Graham, and W.M. Kantor, The Mathematics of perfect shuffles, Adv. in Applied Math. 4, 175-196 (1983).