# weak

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## 6 matching pages

##### 2: 26.14 Permutations: Order Notation
A weak excedance is a position $j$ for which $\sigma(j)\geq j$. …It is also equal to the number of permutations in $\mathfrak{S}_{n}$ with exactly $k+1$ weak excedances. …
##### 3: 14.19 Toroidal (or Ring) Functions
14.19.2 $P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(\frac{1}{2}-% \mu\right)}{\pi^{1/2}\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*% \mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu;1-e^{-2\xi}\right),$ $\mu\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots$.
##### 4: Bibliography
• G. E. Andrews (1979) Plane partitions. III. The weak Macdonald conjecture. Invent. Math. 53 (3), pp. 193–225.
• ##### 5: 26.12 Plane Partitions
A strict shifted plane partition is an arrangement of the parts in a partition so that each row is indented one space from the previous row and there is weak decrease across rows and strict decrease down columns. …
##### 6: Errata
• Equation (14.19.2)
14.19.2 $P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(\frac{1}{2}-% \mu\right)}{{\pi}^{1/2}\left(1-{\mathrm{e}}^{-2\xi}\right)^{\mu}{\mathrm{e}}^{% (\nu+(1/2))\xi}}\*\mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu% ;1-{\mathrm{e}}^{-2\xi}\right),$ $\mu\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots$

Originally the argument to $\mathbf{F}$ in this equation was incorrect (${\mathrm{e}}^{-2\xi}$, rather than $1-{\mathrm{e}}^{-2\xi}$), and the condition on $\mu$ was too weak ($\mu\neq\frac{1}{2}$, rather than $\mu\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots$). Also, the factor multiplying $\mathbf{F}$ was rewritten to clarify the poles; originally it was $\frac{\Gamma\left(1-2\mu\right)2^{2\mu}}{\Gamma\left(1-\mu\right)\left(1-{% \mathrm{e}}^{-2\xi}\right)^{\mu}{\mathrm{e}}^{(\nu+(1/2))\xi}}$.

Reported 2010-11-02 by Alvaro Valenzuela.