How can I convert sequences to sharings and vice versa?

Question

Given positive integers $k,n$, a $k$-sequence of $I_n$ is a list of $k$ not necessarily distinct elements of $\{1,\dots, n\}$. And an $n$-sharing of $I_k$ is a list of $n$ possibly empty, disjoint subsets of $\{1,\dots, k\}$ that union up to $\{1,\dots, k\}$. There is a one-to-one correspondence between the $k$-sequences of $I_n$ and the $n$-sharings of $I_k$, where the $i$th element of the sharing contains all indices $j$ of the sequence that store the number $i$. For example, the $3$-sequence $$(4, 1, 1)$$ of $I_4$ is mapped to the $4$-sharing $$(\{2,3\}, \varnothing, \varnothing, \{1\})$$ of $I_3$ because the two indices storing a $1$ in the sequence are $2$ and $3$, the only index in the sequence storing a $4$ is $1$, and there are no indices storing $2$ or $3$. How can I convert between $k-$sequences of $I_n$ and $n$-sharings of $I_k$ in mathematica?

Answer 1

Looks like a case for PositionIndex:

ClearAll[toPositions]
toPositions = PositionIndex[#] /@ Range[Max@#] /. _Missing -> {} &;

Examples:

toPositions @ {4, 1, 1}

{{2, 3}, {}, {}, {1}}

toPositions @ {5, 2, 3, 2, 5, 3, 5, 5}

{{}, {2, 4}, {3, 6}, {}, {1, 5, 7, 8}}

To recover the input list from positions:

ClearAll[fromPositions]
fromPositions = Values @* KeySort @* 
  KeyValueMap[Sequence @@ Thread[List /@ # -> First@#2] &]
    @* KeyDrop[{{}}] @* PositionIndex;

Examples:

fromPositions@{{2, 3}, {}, {}, {1}}

 {4, 1, 1}

fromPositions@{{}, {2, 4}, {3, 6}, {}, {1, 5, 7, 8}}

 {5, 2, 3, 2, 5, 3, 5, 5}

SeedRandom[1]
list = RandomChoice[Range@6, 10]

{5, 3, 5, 1, 2, 1, 1, 3, 1, 1}

toPositions@list

{{4, 6, 7, 9, 10}, {5}, {2, 8}, {}, {1, 3}}

fromPositions @ %

{5, 3, 5, 1, 2, 1, 1, 3, 1, 1}

SeedRandom[1]
lists = RandomChoice[Range@7, {100, 20}];

AllTrue[fromPositions@toPositions@# == # &] @ lists

True