multiset

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►Let

S=\{1^{a_{1}},2^{a_{2}},\ldots,n^{a_{n}}\}

be the multiset that has

a_{j}

copies of

j

1\leq j\leq n

. … ►The definitions of inversion number and major index can be extended to permutations of a multiset such as

351322453154\in\mathfrak{S}_{\{1^{2},2^{2},3^{3},4^{2},5^{3}\}}

. … ►

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},

ⓘ

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

►

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}.

ⓘ

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.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §26.16 and Ch.26