The fiber product at work (in elementary combinatorics)

In the post “The bimodule of sets, functions and permutations, and its quotients”
we have seen (that $\SetXY$ generates a double quotient square diagram),
and, when $\source{(\setX=[4]=\{0,1,2,3\})}$ and $\target{(\setY=\{\elta,\eltb\})}$,
(the classifications of (the $2^4=16$ elements of $\SetXY$) according to their $\symsX$-orbits or $\symtY$-orbits).

The comparison map into (actually, onto) the fibre product, i.e., pullback,
$\SetXY \longrightarrow \source(\symsX \source\backslash \SetXY\source) \times_{(\symsX \source\backslash \SetXY \target/ \symtY)} \target(\SetXY \target/ \symtY\target)$
when $\source{(\setX=[4]=\{0,1,2,3\})}$ and $\target{(\setY=\{\elta,\eltb\})}$

$\SetXY \rlap{\target{{} \xrightarrow[\mkern42em]{\textstyle \functiont} {}}}$							$\mkern-.5em \SetXY \target/ \symtY$
	$\displaystyle { \source{0123 \mid {} } \brack \target{aaaa} }$	$\displaystyle { \source{ {} \mid 0123} \brack \target{bbbb} }$					$0123\mid{}$		${\cdot} {\cdot} {\cdot} {\cdot}$ $4+0$		$\displaystyle {4 \brace 1}$
$\smash{\llap{\functions}\Bigg\downarrow} \smash{\Rule {1px} {30ex} {25ex}}$			$\displaystyle { \source{012 \mid 3} \brack \target{aaab} }$	$\displaystyle { \source{3 \mid 012} \brack \target{bbba} }$			$012\mid3$		$\begin{subarray}{l} {\cdot} {\cdot} {\cdot}\\ {\cdot}\\ \end{subarray}$ $3+1$		$\displaystyle {4 \brace 2}$
			$\displaystyle { \source{013 \mid 2} \brack \target{aaba} }$	$\displaystyle { \source{2 \mid 013} \brack \target{bbab} }$			$013\mid2$
			$\displaystyle { \source{023 \mid 1} \brack \target{abaa} }$	$\displaystyle { \source{1 \mid 023} \brack \target{babb} }$			$023\mid1$
			$\displaystyle { \source{123 \mid 0} \brack \target{baaa} }$	$\displaystyle { \source{0 \mid 123} \brack \target{abbb} }$			$123\mid0$
					$\displaystyle { \source{01 \mid 23} \brack \target{aabb} }, { \source{23 \mid 01} \brack \target{bbaa} }$		$01\mid23$		$\begin{subarray}{l} {\cdot} {\cdot} \\ {\cdot} {\cdot} \\ \end{subarray}$ $2+2$
					$\displaystyle { \source{02 \mid 13} \brack \target{abab} }, { \source{13 \mid 02} \brack \target{baba} }$		$02\mid13$
$\curlyvee$					$\displaystyle { \source{03 \mid 12} \brack \target{abba} },{ \source{12 \mid 03} \brack \target{baab} }$		$03\mid12$
$\smash{\begin{array}{} \symsX \source\backslash \SetXY\\ \displaystyle \bigg(\mkern-.35em {\target{{\setY=\{\elta,\eltb\}}} \choose \source{|\setX|=4}} \mkern-.35em\bigg)\\ \end{array}}$	${\cdot} {\cdot} {\cdot} {\cdot} \mid {}$ $4+0$ $\elta^4$	${} \mid {\cdot} {\cdot} {\cdot} {\cdot}$ $0+4$ $\eltb^4$	${\cdot} {\cdot} {\cdot} \mid \cdot$ $3+1$ $\elta^3\eltb$	$\cdot \mid {\cdot} {\cdot} {\cdot}$ $1+3$ $\elta\eltb^3$	${\cdot \cdot} \mid {\cdot \cdot}$ $2+2$ $\elta^2\eltb^2$
	${\cdot} {\cdot} {\cdot} {\cdot}$ $4+0$		$\begin{subarray}{l} {\cdot} {\cdot} {\cdot}\\ {\cdot}\\ \end{subarray}$ $3+1$		$\begin{subarray}{l} {\cdot} {\cdot} \\ {\cdot} {\cdot} \\ \end{subarray}$ $2+2$				$\mkern-2em \symsX \source\backslash \SetXY \target/ \symtY$
$\big( \setY = \{\elta,\eltb\} \big) \calP^{{+}}$	${\cdot}\mid{}$ $\{\elta\}$	${}\mid{\cdot}$ $\{\eltb\}$	${\cdot}\mid{\cdot}$ $\{\elta,\eltb\}$
	$\displaystyle {2 \choose 1}1!$		$\displaystyle {2 \choose 2}2!$								$\text{image}$ $\text{size}$ $1 \text{ or } 2$

The comparison map at right
combines all three aspects.

Here the thick edges bound
1) the three elements in the double quotient (d.q.), and
2) fibres of (those three d.q. elements) with respect to any of (the three maps into (actually, onto) the d.q.).

The thin edges separate
1) elements in the two single quotients, and
2) fibres of (those elements) w.r.t. either of (the two maps $\functions,\functiont$ from $\SetXY$ onto the two single quotients):
(the thin horizontal lines) separate
(the fibres over $\SetXY \target{/\symtY}$),
i.e., (the $\symtY$-orbits $?\symtY$);
(the thin vertical lines) separate
(the fibres over $\source{\symsX\backslash}\SetXY$),
i.e., (the $\symsX$-orbits $?\symsX$).
(The fiber in $\SetXY$ over $2+2$)
has no vertical line,
(its six functions) are all in
(the same $\symsX$-orbit).

The dashed cells correspond to other decompositions of $\SetXY$,
according either to (the images),
or to (the size, $1$ or $2$, of the images),
thus also to
(the size (number of blocks) of
(the kernel-partitions)).
The contents of some of the dashed cells are the factors that appear in the sum $${\target{|\setY|}}^{\source{|\setX|}} = \displaystyle\target{\sum_{i=0}^{|\setY|}} \source{{|\setX| \brace \target i}}\target{i!}\target{{|\setY| \choose i}}$$ when ($\source{\setX=[4]}$), ($\target{\setY=\{\elta,\eltb\}}$),
and ($i=1$ or $2$) —
thus demonstrating, in this case,
the validity of that equality.

The above exhibits four (among many) ways of partitioning/decomposing/classifying $\SetXY$:
orbits under $\symsX$, orbits under $\symtY$, orbits under $\symsX\op\times\symtY$, and by (the size of the image).

This, among other things, provides an example of a two-way classification scheme.

An alternative description of the cells:

Here the thick edges bound
1) the three elements of the double quotient (d.q.), that is,
(the three orbits $\symsX?\symtY$ in $\SetXY$) of (the combined (product) group $\symsX\op\times\symtY$),
containing (two, eight, and six functions),
2) (the unordered $2$-partitions of $4$, $4+0$, $3+1$, $2+2$) associated with, and designating, (those orbits), and
3) the intermediate subsets of (the designators for the single quotients).

The thin edges separate
1) the orbits of the single group actions, that is, of $\symsX\op$ and $\symtY$ in $\SetXY$, and
2) the designators for those orbits as
a) unordered $2$-partitions of $[4]$ (for the $\symtY$ action) and
b) ordered $2$-partitions of $4$ (for the $\symsX$ action).

$\homst {[2]} \Set {\{\elta,\eltb\}}$	$\homst {[2]} \Set {\{\elta,\eltb,\eltc\}}$	$\homst {[2]} \Set {\{\elta,\eltb,\eltc,\eltd\}}$
$\homst {[3]} \Set {\{\elta,\eltb\}}$	$\homst {[3]} \Set {\{\elta,\eltb,\eltc\}}$
$\homst {[4]} \Set {\{\elta,\eltb\}}$

The table at right lists some (clickable) further examples, given below:

---

The comparison map into (actually, onto) the fibre product, i.e., pullback,
$\SetXY \longrightarrow \source(\symsX \source\backslash \SetXY\source) \times_{(\symsX \source\backslash \SetXY \target/ \symtY)} \target(\SetXY \target/ \symtY\target)$
when $\source{(\setX=[2]=\{0,1\})}$ and $\target{(\setY=\{\elta,\eltb\})}$

$\SetXY \rlap{\target{{} \xrightarrow[\mkern23em]{\textstyle \functiont} {}}}$						$\mkern-1em \SetXY \target/ \symtY$
$\smash{\llap{\functions}\Bigg\downarrow} \smash{\Rule {1px} {10ex} {7.5ex}}$		$\displaystyle { \source{01 \mid {} } \brack \target{aa} }$	$\displaystyle { \source{ {} \mid 01} \brack \target{bb} }$			$01\mid{}$		${\cdot} {\cdot}$ $2+0$		$\displaystyle {2 \brace 1}$
	v			$\displaystyle { \source{0 \mid 1} \brack \target{ab} }, { \source{1 \mid 0} \brack \target{ba} }$		$0\mid1$		$\begin{subarray}{l} {\cdot} \\ {\cdot}\\ \end{subarray}$ $1+1$		$\displaystyle {2 \brace 2}$
$\smash{\begin{array}{} \symsX \source\backslash \SetXY\\ \displaystyle \bigg(\mkern-.35em {\target{{\setY=\{\elta,\eltb\}}} \choose \source{|\setX|=2}} \mkern-.35em\bigg)\\ \end{array}}$		${\cdot} {\cdot} \mid {}$ $2+0$ $\elta^2$	${} \mid {\cdot} {\cdot}$ $0+2$ $\eltb^2$	${\cdot} \mid {\cdot}$ $1+1$ $\elta\eltb$
		${\cdot} {\cdot}$ $2+0$		$\begin{subarray}{l} {\cdot} \\ {\cdot}\\ \end{subarray}$ $1+1$				$\mkern-1.5em \symsX \source\backslash \SetXY \target/ \symtY$
$\big( \setY = \{\elta,\eltb\} \big) \calP^{{+}}$		${\cdot}\mid{}$ $\{\elta\}$	${}\mid{\cdot}$ $\{\eltb\}$	${\cdot}\mid{\cdot}$ $\{\elta,\eltb\}$
		$\displaystyle {2 \choose 1}1!$		$\displaystyle {2 \choose 2}2!$						$\text{image}$ $\text{size}$ $1 \text{ or } 2$

The comparison map into (actually, onto) the fibre product, i.e., pullback,
$\SetXY \longrightarrow \source(\symsX \source\backslash \SetXY\source) \times_{(\symsX \source\backslash \SetXY \target/ \symtY)} \target(\SetXY \target/ \symtY\target)$
when $\source{(\setX=[2]=\{0,1\})}$ and $\target{(\setY=\{\elta,\eltb,\eltc\})}$

$\SetXY \rlap{\target{{} \xrightarrow[\mkern45em]{\textstyle \functiont} {}}}$

$\mkern-.5em \SetXY \target/ \symtY$

$\displaystyle { \source{01 \mid \mid {} } \brack \target{aa} }$

$\displaystyle { \source{ {} \mid 01 \mid {}} \brack \target{bb} }$

$\displaystyle { \source{ {} \mid \mid 01} \brack \target{cc} }$

$01\mid\mid$

${\cdot} {\cdot}$ $2+0+0$

$\displaystyle {2 \brace 1}$

$\smash{\llap{\functions}\Bigg\downarrow} \smash{\Rule {1px} {12ex} {9ex}}$

$\displaystyle { \source{0 \mid 1 \mid {}} \brack \target{\elta\eltb} }$

$\displaystyle { \source{0 \mid\mid 1} \brack \target{\elta\eltc} }$

$\displaystyle { \source{{} \mid 0 \mid 1 } \brack \target{\eltb\eltc} }$

$0\mid1\mid{}$

$\begin{subarray}{l} {\cdot}\\ {\cdot}\\ \end{subarray}$ $1+1+0$

$\displaystyle {2 \brace 2}$

v

$\displaystyle { \source{1 \mid 0 \mid{} } \brack \target{\eltb\elta} }$

$\displaystyle { \source{1 \mid\mid 0 } \brack \target{\eltc\elta} }$

$\displaystyle { \source{{}\mid1\mid0} \brack \target{\eltc\eltb} }$

$\smash{\begin{array}{} \symsX \source\backslash \SetXY\\ \displaystyle \bigg(\mkern-.35em {\target{{\setY=\{\elta,\eltb,\eltc\}}} \choose \source{|\setX|=2}} \mkern-.35em\bigg)\\ \end{array}}$

${\cdot} {\cdot} \mid\mid {}$ $2+0+0$ $\elta^2$

${} \mid {\cdot} {\cdot} \mid {}$ $0+2+0$ $\eltb^2$

${} \mid\mid {\cdot} {\cdot}$ $0+0+2$ $\eltc^2$

${\cdot} \mid {\cdot} \mid {}$ $1+1+0$ $\elta\eltb$

${\cdot} \mid\mid {\cdot}$ $1+0+1$ $\elta\eltc$

${} \mid {\cdot} \mid {\cdot}$ $0+1+1$ $\eltb\eltc$

${\cdot} {\cdot}$ $2+0+0$

$\begin{subarray}{l} {\cdot}\\ {\cdot}\\ \end{subarray}$ $1+1+0$

$\mkern-1em \symsX \source\backslash \SetXY \target/ \symtY$

$\big( \setY = \{\elta,\eltb,\eltc\} \big) \calP_{1\leq 2}$

${\cdot} \mid\mid {}$ $\{\elta\}$

${} \mid {\cdot} \mid {}$ $\{\eltb\}$

${} \mid\mid {\cdot}$ $\{\eltc\}$

${\cdot} \mid {\cdot} \mid {}$ $\{\elta,\eltb\}$

${\cdot} \mid \mid {\cdot}$ $\{\elta,\eltc\}$

${} \mid {\cdot} \mid {\cdot}$ $\{\eltb,\eltc\}$

$\displaystyle {3 \choose 1}1!$

$\displaystyle {3 \choose 2}2!$

$\text{image}$ $\text{size}$ $1 \text{ or } 2$

---

The comparison map into (actually, onto) the fibre product, i.e., pullback,
$\SetXY \longrightarrow \source(\symsX \source\backslash \SetXY\source) \times_{(\symsX \source\backslash \SetXY \target/ \symtY)} \target(\SetXY \target/ \symtY\target)$
when $\source{(\setX=[2]=\{0,1\})}$ and $\target{(\setY=\{\elta,\eltb,\eltc,\eltd \})}$

$\SetXY \rlap{\target{{} \xrightarrow[\mkern92em]{\textstyle \functiont} {}}}$

$\mkern-.5em \SetXY \target/ \symtY$

$\displaystyle { \source{01 \mid \mid \mid {} } \brack \target{aa} }$

$\displaystyle { \source{ {} \mid 01 \mid \mid {}} \brack \target{bb} }$

$\displaystyle { \source{ {} \mid \mid 01 \mid {}} \brack \target{cc} }$

$\displaystyle { \source{ {} \mid \mid \mid 01} \brack \target{dd} }$

$01\mid\mid\mid$

${\cdot} {\cdot}$ $2+0+0+0$

$\displaystyle {2 \brace 1}$

$\smash{\llap{\functions}\Bigg\downarrow} \smash{\Rule {1px} {12ex} {9ex}}$

$\displaystyle { \source{0 \mid 1 \mid \mid {}} \brack \target{\elta\eltb} }$

$\displaystyle { \source{0 \mid \mid 1 \mid {}} \brack \target{\elta\eltc} }$

$\displaystyle { \source{0 \mid \mid \mid 1} \brack \target{\elta\eltd} }$

$\displaystyle { \source{{} \mid 0 \mid 1 \mid {} } \brack \target{\eltb\eltc} }$

$\displaystyle { \source{{} \mid 0 \mid \mid 1 } \brack \target{\eltb\eltd} }$

$\displaystyle { \source{{} \mid \mid 0 \mid 1 } \brack \target{\eltc\eltd} }$

$0\mid1\mid\mid{}$

$\begin{subarray}{l} {\cdot}\\ {\cdot}\\ \end{subarray}$ $1+1+0+0$

$\displaystyle {2 \brace 2}$

v

$\displaystyle { \source{1 \mid 0 \mid\mid {} } \brack \target{\eltb\elta} }$

$\displaystyle { \source{1 \mid\mid 0 \mid {} } \brack \target{\eltc\elta} }$

$\displaystyle { \source{1 \mid\mid\mid 0 } \brack \target{\eltd\elta} }$

$\displaystyle { \source{{}\mid1\mid0\mid {}} \brack \target{\eltc\eltb} }$

$\displaystyle { \source{{}\mid1\mid\mid 0} \brack \target{\eltd\eltb} }$

$\displaystyle { \source{{}\mid\mid1\mid0} \brack \target{\eltd\eltc} }$

$\smash{\begin{array}{} \symsX \source\backslash \SetXY\\ \displaystyle \bigg(\mkern-.35em {\target{{\setY=\{\elta,\eltb,\eltc,\eltd\}}} \choose \source{|\setX|=2}} \mkern-.35em\bigg)\\ \end{array}}$

${\cdot} {\cdot} \mid\mid\mid {}$ $2+0+0+0$ $\elta^2$

${} \mid {\cdot} {\cdot} \mid\mid {}$ $0+2+0+0$ $\eltb^2$

${} \mid\mid {\cdot} {\cdot} \mid{}$ $0+0+2+0$ $\eltc^2$

${} \mid\mid\mid {\cdot} {\cdot}$ $0+0+0+2$ $\eltd^2$

${\cdot} \mid {\cdot} \mid\mid {}$ $1+1+0+0$ $\elta\eltb$

${\cdot} \mid\mid {\cdot}\mid {}$ $1+0+1+0$ $\elta\eltc$

${\cdot} \mid\mid\mid {\cdot}$ $1+0+0+1$ $\elta\eltd$

${}\mid {\cdot} \mid {\cdot} \mid{}$ $0+1+1+0$ $\eltb\eltc$

${}\mid {\cdot} \mid\mid {\cdot}$ $0+1+0+1$ $\eltb\eltd$

${} \mid\mid {\cdot} \mid {\cdot}$ $0+0+1+1$ $\eltc\eltd$

${\cdot} {\cdot}$ $2+0+0+0$

$\begin{subarray}{l} {\cdot}\\ {\cdot}\\ \end{subarray}$ $1+1+0+0$

$\mkern-1em \symsX \source\backslash \SetXY \target/ \symtY$

$\big( \setY = \{\elta,\eltb,\eltc\} \big) \calP_{1\leq 2}$

${\cdot} \mid\mid\mid {}$ $\{\elta\}$

${} \mid {\cdot} \mid\mid {}$ $\{\eltb\}$

${} \mid\mid {\cdot} \mid {}$ $\{\eltc\}$

${} \mid\mid\mid {\cdot}$ $\{\eltd\}$

${\cdot} \mid {\cdot} \mid \mid {}$ $\{\elta,\eltb\}$

${\cdot} \mid \mid {\cdot} \mid {}$ $\{\elta,\eltc\}$

${\cdot} \mid \mid \mid {\cdot}$ $\{\elta,\eltd\}$

${}\mid {\cdot} \mid {\cdot} \mid {}$ $\{\eltb,\eltc\}$

${}\mid {\cdot} \mid \mid {\cdot}$ $\{\eltb,\eltd\}$

${} \mid\mid {\cdot} \mid {\cdot}$ $\{\eltc,\eltd\}$

$\displaystyle {4 \choose 1}1!$

$\displaystyle {4 \choose 2}2!$

$\text{image}$ $\text{size}$ $1 \text{ or } 2$

---

The comparison map into (actually, onto) the fibre product, i.e., pullback,
$\SetXY \longrightarrow \source(\symsX \source\backslash \SetXY\source) \times_{(\symsX \source\backslash \SetXY \target/ \symtY)} \target(\SetXY \target/ \symtY\target)$
when $\source{(\setX=[3]=\{0,1,2\})}$ and $\target{(\setY=\{\elta,\eltb\})}$

$\SetXY \rlap{\target{{} \xrightarrow[\mkern25em]{\textstyle \functiont} {}}}$						$\mkern-1em \SetXY \target/ \symtY$
	$\displaystyle { \source{012 \mid {} } \brack \target{aaa} }$	$\displaystyle { \source{ {} \mid 012} \brack \target{bbb} }$				$012\mid{}$		${\cdot} {\cdot} {\cdot}$ $3+0$		$\displaystyle {3 \brace 1}$
$\smash{\llap{\functions}\Bigg\downarrow} \smash{\Rule {1px} {15ex} {10.5ex}}$			$\displaystyle { \source{01 \mid 2} \brack \target{aab} }$	$\displaystyle { \source{2 \mid 01} \brack \target{bba} }$		$01\mid2$		$\begin{subarray}{l} {\cdot} {\cdot} \\ {\cdot}\\ \end{subarray}$ $2+1$		$\displaystyle {3 \brace 2}$
			$\displaystyle { \source{02 \mid 1} \brack \target{aba} }$	$\displaystyle { \source{1 \mid 02} \brack \target{bab} }$		$02\mid1$
$\vee$			$\displaystyle { \source{12 \mid 0} \brack \target{baa} }$	$\displaystyle { \source{0 \mid 12} \brack \target{abb} }$		$12\mid0$
$\smash{\begin{array}{} \symsX \source\backslash \SetXY\\ \displaystyle \bigg(\mkern-.35em {\target{{\setY=\{\elta,\eltb\}}} \choose \source{|\setX|=3}} \mkern-.35em\bigg)\\ \end{array}}$	${\cdot} {\cdot} {\cdot} \mid {}$ $3+0$ $\elta^3$	${} \mid {\cdot} {\cdot} {\cdot}$ $0+3$ $\eltb^3$	${\cdot} {\cdot} \mid {\cdot}$ $2+1$ $\elta^2\eltb$	${\cdot} \mid {\cdot} {\cdot}$ $1+2$ $\elta\eltb^2$
	${\cdot} {\cdot} {\cdot}$ $3+0$		$\begin{subarray}{l} {\cdot} {\cdot} \\ {\cdot}\\ \end{subarray}$ $2+1$					$\mkern-1.5em \symsX \source\backslash \SetXY \target/ \symtY$
$\big( \setY = \{\elta,\eltb\} \big) \calP^{{+}}$	${\cdot}\mid{}$ $\{\elta\}$	${}\mid{\cdot}$ $\{\eltb\}$	${\cdot}\mid{\cdot}$ $\{\elta,\eltb\}$
	$\displaystyle {2 \choose 1}1!$		$\displaystyle {2 \choose 2}2!$							$\text{image}$ $\text{size}$ $1 \text{ or } 2$

---

The comparison map into (actually, onto) the fibre product, i.e., pullback,
$\SetXY \longrightarrow \source(\symsX \source\backslash \SetXY\source) \times_{(\symsX \source\backslash \SetXY \target/ \symtY)} \target(\SetXY \target/ \symtY\target)$
when $\source{(\setX=[3]=\{0,1,2\})}$ and $\target{(\setY=\{\elta,\eltb,\eltc\})}$

$\SetXY \rlap{\target{{} \xrightarrow[\mkern77em]{\textstyle \functiont} {}}}$

$\mkern-.5em \SetXY \target/ \symtY$

$\displaystyle { \source{012 \mid \mid {} } \brack \target{aaa} }$

$\displaystyle { \source{ {} \mid 012 \mid {}} \brack \target{bbb} }$

$\displaystyle { \source{ {} \mid \mid 012} \brack \target{ccc} }$

$012\mid\mid$

${\cdot} {\cdot} {\cdot}$ $3+0+0$

$\displaystyle {3 \brace 1}$

$\smash{\llap{\functions}\Bigg\downarrow} \smash{\Rule {1px} {20ex} {27ex}}$

$\displaystyle { \source{01 \mid 2 \mid {}} \brack \target{\elta\elta\eltb} }$

$\displaystyle { \source{2\mid 01\mid{}} \brack \target{\eltb\eltb\elta} }$

$\displaystyle { \source{01 \mid\mid 2} \brack \target{\elta\elta\eltc} }$

$\displaystyle { \source{2\mid\mid 01} \brack \target{\eltc\eltc\elta} }$

$\displaystyle { \source{{} \mid 01\mid 2} \brack \target{\eltb\eltb\eltc} }$

$\displaystyle { \source{{} \mid 2 \mid 01} \brack \target{\eltc\eltc\eltb} }$

$01\mid2\mid{}$

$\begin{subarray}{l} {\cdot} {\cdot}\\ {\cdot}\\ \end{subarray}$ $2+1+0$

$\displaystyle {3 \brace 2}$

$\displaystyle { \source{02 \mid 1 \mid{} } \brack \target{\elta\eltb\elta} }$

$\displaystyle { \source{1\mid02\mid{}} \brack \target{\eltb\elta\eltb} }$

$\displaystyle { \source{02 \mid\mid 1 } \brack \target{\elta\eltc\elta} }$

$\displaystyle { \source{1\mid\mid02} \brack \target{\eltc\elta\eltc} }$

$\displaystyle { \source{{}\mid02\mid1} \brack \target{\eltb\eltc\eltb} }$

$\displaystyle { \source{{}\mid1\mid02} \brack \target{\eltc\eltb\eltc} }$

$02\mid1\mid{}$

$\displaystyle { \source{12 \mid 0 \mid{} } \brack \target{\eltb\elta\elta} }$

$\displaystyle { \source{0\mid12\mid{}} \brack \target{\elta\eltb\eltb} }$

$\displaystyle { \source{12 \mid\mid0} \brack \target{\eltc\elta\elta} }$

$\displaystyle { \source{0\mid\mid12} \brack \target{\elta\eltc\eltc} }$

$\displaystyle { \source{{}\mid12\mid0} \brack \target{\eltc\eltb\eltb} }$

$\displaystyle { \source{{}\mid0\mid12} \brack \target{\eltb\eltc\eltc} }$

$12\mid0\mid{}$

v

$\displaystyle {\source{0\mid1\mid2} \brack \target{\elta\eltb\eltc}}, \displaystyle {\source{0\mid2\mid1} \brack \target{\elta\eltc\eltb}},$ $\displaystyle {\source{1\mid0\mid2} \brack \target{\eltb\elta\eltc}}, \displaystyle {\source{2\mid0\mid1} \brack \target{\eltb\eltc\elta}},$ $\displaystyle {\source{1\mid2\mid0} \brack \target{\eltc\elta\eltb}}, \displaystyle {\source{2\mid1\mid0} \brack \target{\eltc\eltb\elta}}\;$

$0\mid1\mid2$

$\begin{subarray}{l} {\cdot}\\ {\cdot}\\ {\cdot}\\ \end{subarray}$ $1+1+1$

$\displaystyle {3 \brace 3}$

$\smash{\begin{array}{} \symsX \source\backslash \SetXY\\ \displaystyle \bigg(\mkern-.35em {\target{{\setY=\{\elta,\eltb,\eltc\}}} \choose \source{|\setX|=3}} \mkern-.35em\bigg)\\ \end{array}}$

${\cdot} {\cdot} {\cdot} \mid\mid{}$ $3+0+0$ $\elta^3$

${}\mid {\cdot} {\cdot} {\cdot} \mid{}$ $0+3+0$ $\eltb^3$

${}\mid\mid {\cdot} {\cdot} {\cdot}$ $0+0+3$ $\eltc^3$

${\cdot} {\cdot} \mid {\cdot}\mid{}$ $2+1+0$ $\elta^2\eltb$

${\cdot} \mid {\cdot} {\cdot}\mid{}$ $1+2+0$ $\elta\eltb^2$

${\cdot} {\cdot} \mid\mid {\cdot}$ $2+0+1$ $\elta^2\eltc$

${\cdot} \mid\mid {\cdot} {\cdot}$ $1+0+2$ $\elta\eltc^2$

${}\mid {\cdot} {\cdot} \mid {\cdot}$ $0+2+1$ $\eltb^2\eltc$

${}\mid {\cdot} \mid {\cdot} {\cdot}$ $0+1+2$ $\eltb\eltc^2$

${\cdot} \mid {\cdot} \mid {\cdot}$ $1+1+1$ $\elta\eltb\eltc$

${\cdot} {\cdot} {\cdot}$ $3+0+0$

$\begin{subarray}{l} {\cdot} {\cdot}\\ {\cdot}\\ \end{subarray}$ $2+1+0$

$\begin{subarray}{l} {\cdot}\\ {\cdot}\\ {\cdot}\\ \end{subarray}$ $1+1+1$

$\mkern-1em \symsX \source\backslash \SetXY \target/ \symtY$

$\big( \setY = \{\elta,\eltb,\eltc\} \big) \calP^{{+}}$

${\cdot} \mid\mid {}$ $\{\elta\}$

${} \mid {\cdot} \mid {}$ $\{\eltb\}$

${} \mid\mid {\cdot}$ $\{\eltc\}$

${\cdot} \mid {\cdot} \mid {}$ $\{\elta,\eltb\}$

${\cdot} \mid \mid {\cdot}$ $\{\elta,\eltc\}$

${} \mid {\cdot} \mid {\cdot}$ $\{\eltb,\eltc\}$

${\cdot} \mid {\cdot} \mid {\cdot}$ $\{\elta,\eltb,\eltc\}$

$\displaystyle {3 \choose 1}1!$

$\displaystyle {3 \choose 2}2!$

$\displaystyle {3 \choose 3}3!$

$\text{image}$ $\text{size}$ $1, 2, \text{ or } 3$

---

References

EC, Stanley, Richard P., Enumerative Combinatorics I, 1986/2011
W, Wikipedia, “Twelvefold way”
BMSFP, “The bimodule of sets, functions and permutations, and its quotients”, 2019
DG12W, “Double groupoids and the twelvefold way”, 2016
PF, “The parts of a function”, 2016
CFP, “Classifying functions by their parts”, 2016
QKA, “The quotient-kernel adjunction”, 2016
KQFC, “Kernels, quotients, and function composition”, 2019

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