multinomial coefficients

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Table 26.4.1: Multinomials and partitions.

$n$	$m$	$\lambda$	$M_{1}$	$M_{2}$	$M_{3}$
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… ►The number of elements in

\mathfrak{S}_{S}

is the multinomial coefficient (§26.4)

\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},

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Symbols:: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $\genfrac{[}{]}{0.0pt}{}{\NVar{a_{1}}+\NVar{a_{2}}+\dots+\NVar{a_{n}}}{\NVar{a_% {1}},\NVar{a_{2}},\ldots,\NVar{a_{n}}}_{\NVar{q}}$ : $q$ -multinomial coefficient, $k$ : nonnegative integer, $n$ : nonnegative integer, $a_{j}$ : numbers and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.16 and Ch.26

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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},

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Symbols:: $\in$ : element of, $\genfrac{[}{]}{0.0pt}{}{\NVar{a_{1}}+\NVar{a_{2}}+\dots+\NVar{a_{n}}}{\NVar{a_% {1}},\NVar{a_{2}},\ldots,\NVar{a_{n}}}_{\NVar{q}}$ : $q$ -multinomial coefficient, $\mathfrak{S}_{\NVar{n}}$ : set of permutations of $\{1,2,\ldots,n\}$ , $n$ : nonnegative integer, $a_{j}$ : numbers, $S$ : multiset and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.16.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §26.16 and Ch.26

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… ►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!},

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : hyperbolic umbilic canonical integral function, $\mathrm{i}$ : imaginary unit, $y$ : real parameter, $z$ : real parameter, $m$ : integer, $x$ : real parameter, $c_{n}(t)$ : coefficients and $d_{n}(t)$ : coefficients
Referenced by:: §36.8
Permalink:: http://dlmf.nist.gov/36.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §36.8 and Ch.36

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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),

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Defines:: $f_{n}$ : function (locally)
Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\overline{\NVar{z}}$ : complex conjugate, $m$ : integer, $c_{n}(t)$ : coefficients and $d_{n}(t)$ : coefficients
Permalink:: http://dlmf.nist.gov/36.8.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §36.8 and Ch.36

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