# q-multinomial coefficient

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##### 1: 26.16 Multiset Permutations
Thus $\mathop{\mathrm{inv}}(351322453154)=4+8+0+3+1+1+2+3+1+0+1=24$, and $\mathop{\mathrm{maj}}(351322453154)=2+4+8+9+11=34.$ The $q$-multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by
26.16.1 $\genfrac{[}{]}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}_{q% }=\prod_{k=1}^{n-1}\genfrac{[}{]}{0.0pt}{}{a_{k}+a_{k+1}+\cdots+a_{n}}{a_{k}}_% {q},$
26.16.2 $\sum_{\sigma\in\mathfrak{S}_{S}}q^{\mathop{\mathrm{inv}}(\sigma)}=\genfrac{[}{% ]}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}_{q},$
26.16.3 $\sum_{\sigma\in\mathfrak{S}_{S}}q^{\mathop{\mathrm{maj}}(\sigma)}=\genfrac{[}{% ]}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}_{q}.$
##### 3: 26.21 Tables
###### §26.21 Tables
Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\genfrac{(}{)}{0.0pt}{}{m}{n}$ for $m$ up to 50 and $n$ up to 25; extends Table 26.4.1 to $n=10$; tabulates Stirling numbers of the first and second kinds, $s\left(n,k\right)$ and $S\left(n,k\right)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p\left(\mathcal{D},n\right)$ for $n$ up to 500. … Goldberg et al. (1976) contains tables of binomial coefficients to $n=100$ and Stirling numbers to $n=40$.
##### 4: 28.14 Fourier Series
The coefficients satisfy
28.14.5 $\sum_{m=-\infty}^{\infty}\left(c_{2m}^{\nu}(q)\right)^{2}=1;$
##### 5: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
###### §26.4(i) Definitions
$M_{1}$ is the multinominal coefficient (26.4.2): …
##### 6: 29.20 Methods of Computation
Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. … A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). …The approximations converge geometrically (§3.8(i)) to the eigenvalues and coefficients of Lamé functions as $n\to\infty$. …
###### §29.20(ii) Lamé Polynomials
The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …
##### 7: 15.7 Continued Fractions
15.7.1 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=t_{0% }-\cfrac{u_{1}z}{t_{1}-\cfrac{u_{2}z}{t_{2}-\cfrac{u_{3}z}{t_{3}-\cdots}}},$
where …
15.7.3 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=v_{0% }-\cfrac{w_{1}}{v_{1}-\cfrac{w_{2}}{v_{2}-\cfrac{w_{3}}{v_{3}-\cdots}}},$
where …
##### 9: 29.21 Tables
• Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^{2}=0.1(.1)0.9$, $n=1(1)30$. Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.

• ##### 10: 34.1 Special Notation
An often used alternative to the $\mathit{3j}$ symbol is the Clebsch–Gordan coefficient
34.1.1 $\left(j_{1}\;m_{1}\;j_{2}\;m_{2}|j_{1}\;j_{2}\;j_{3}\,\,m_{3}\right)=(-1)^{j_{% 1}-j_{2}+m_{3}}(2j_{3}+1)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&-m_{3}\end{pmatrix};$
For other notations for $\mathit{3j}$, $\mathit{6j}$, $\mathit{9j}$ symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).