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The algebra and combinatorics of shuffles and multiple zeta values. (English) Zbl 1021.11026
The authors consider multiple zeta values $\zeta (s_1,\ldots ,s_k)=\sum_{n_1>n_2>\ldots >n_k>0} \prod_{j=1}^kn_j^{-s_j}$ where $$s_1,\ldots ,s_k$$ are positive integers, $$s_1>1$$.
In particular, given a vector $$\vec s=(m_0,m_1,\ldots ,m_{2n})$$ of non-negative integers, they study $Z(\vec s)=\zeta \left( \{2\}^{m_0},3,\{2\}^{m_1},1,\{2\}^{m_2},3,\{2\}^{m_3},1,\ldots ,3,\{2\}^{m_{2n-1}},1,\{2\}^{m_{2n}}\right)$ where $$\{2\}^m$$ means the argument 2 repeated $$m$$ times. Sums of $$Z(\vec s)$$ over some sets of $$\vec s$$ are found explicitly, generalizing the formula for $$\zeta (\{ 3,1\}^n)$$ conjectured by D. Zagier and proved by J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisonek [Trans. Am. Math. Soc. 353, 907-941 (2001; Zbl 1002.11093)].
The technique is based on further development of the algebraic and combinatorial theory of shuffles introduced by K.-T. Chen [Proc. Lond. Math. Soc. (3) 4, 502-512 (1954; Zbl 0058.25603)] and R. Ree [Ann. Math. (2) 68, 210-220 (1958; Zbl 0083.25401)].

##### MSC:
 11M41 Other Dirichlet series and zeta functions 05A19 Combinatorial identities, bijective combinatorics 11B75 Other combinatorial number theory
##### Keywords:
shuffle; multiple zeta values; cyclic sum
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##### References:
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