# multinomial coefficients

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##### 2: 26.16 Multiset Permutations
The number of elements in $\mathfrak{S}_{S}$ is the multinomial coefficient26.4) $\genfrac{(}{)}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}$. … 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 $\sum_{\sigma\in\mathfrak{S}_{S}}q^{\mathop{\mathrm{inv}}(\sigma)}=\genfrac{[}{% ]}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}_{q},$
##### 3: 26.1 Special Notation
 $\genfrac{(}{)}{0.0pt}{}{m}{n}$ binomial coefficient. multinomial coefficient. …
##### 4: 36.8 Convergent Series Expansions
For multinomial power series for $\Psi_{K}\left(\mathbf{x}\right)$, see Connor and Curtis (1982).
36.8.3 $\dfrac{3^{2/3}}{4\pi^{2}}\Psi^{(\mathrm{H})}\left(3^{1/3}\mathbf{x}\right)=% \operatorname{Ai}\left(x\right)\operatorname{Ai}\left(y\right)\sum\limits_{n=0% }^{\infty}(-3^{-1/3}iz)^{n}\dfrac{c_{n}(x)c_{n}(y)}{n!}+\operatorname{Ai}\left% (x\right)\operatorname{Ai}'\left(y\right)\sum\limits_{n=2}^{\infty}(-3^{-1/3}% iz)^{n}\dfrac{c_{n}(x)d_{n}(y)}{n!}+\operatorname{Ai}'\left(x\right)% \operatorname{Ai}\left(y\right)\sum\limits_{n=2}^{\infty}(-3^{-1/3}iz)^{n}% \dfrac{d_{n}(x)c_{n}(y)}{n!}+\operatorname{Ai}'\left(x\right)\operatorname{Ai}% '\left(y\right)\sum\limits_{n=1}^{\infty}(-3^{-1/3}iz)^{n}\dfrac{d_{n}(x)d_{n}% (y)}{n!},$
36.8.5 $f_{n}(\zeta,\overline{\zeta})=c_{n}(\zeta)c_{n}(\overline{\zeta})\operatorname% {Ai}\left(\zeta\right)\operatorname{Bi}\left(\overline{\zeta}\right)+c_{n}(% \zeta)d_{n}(\overline{\zeta})\operatorname{Ai}\left(\zeta\right)\operatorname{% Bi}'\left(\overline{\zeta}\right)+d_{n}(\zeta)c_{n}(\overline{\zeta})% \operatorname{Ai}'\left(\zeta\right)\operatorname{Bi}\left(\overline{\zeta}% \right)+d_{n}(\zeta)d_{n}(\overline{\zeta})\operatorname{Ai}'\left(\zeta\right% )\operatorname{Bi}'\left(\overline{\zeta}\right),$