restricted integer partitions

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Table 26.9.1: Partitions

p_{k}\left(n\right)

$n$	$k$
$n$	…

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Figure 26.9.1: Ferrers graph of the partition

7+4+3+3+2+1

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Table 26.10.1: Partitions restricted by difference conditions, or equivalently with parts from

A_{j,k}

	$p\left(\mathcal{D},n\right)$	$p\left(\mathcal{D}2,n\right)$	$p\left(\mathcal{D}2,\hbox{}\!\!\in\!T,n\right)$	$p\left(\mathcal{D}^{\prime}3,n\right)$
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$x$	real variable.
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►A composition is an integer partition in which order is taken into account. …

c\left(n\right)

denotes the number of compositions of

n

, and

c_{m}\left(n\right)

is the number of compositions into exactly

m

parts.

c\left(\in\!T,n\right)

is the number of compositions of

n

with no 1’s, where again

T=\{2,3,4,\ldots\}

. The integer 0 is considered to have one composition consisting of no parts: …