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11: 26.9 Integer Partitions: Restricted Number and Part Size

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26.9.4 $${\left[\genfrac{}{}{0.0pt}{}{m}{n}\right]}_q=\prod _{j=1}^n\frac{1-q^{m-n+j}}{1-q^j},$$ $n\ge 0$,

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

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26.9.5 $$\sum _{n=0}^{\mathrm{\infty }}{p}_k\left(n\right)q^n=\prod _{j=1}^k\frac{1}{1-q^j}=1+\sum _{m=1}^{\mathrm{\infty }}{\left[\genfrac{}{}{0.0pt}{}{k+m-1}{m}\right]}_q{q}^m,$$

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Symbols:

$p_k\left(n\right)$: total number of partitions of $n$ into at most $k$ parts, ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

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26.9.7 $$\sum _{m,n=0}^{\mathrm{\infty }}{p}_k(\le m,n)x^k{q}^n=1+\sum _{k=1}^{\mathrm{\infty }}{\left[\genfrac{}{}{0.0pt}{}{m+k}{k}\right]}_q{x}^k=\prod _{j=0}^m\frac{1}{1-x{q}^j}.$$

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $p_k(\le m,n)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

… ►where the inner sum is taken over all positive divisors of $t$ that are less than or equal to $k$. …