Successor

The element after $x$ in the collection's own order.

✓ checked against that system · ! checked, with disagreements (hover a chip)Combinatorica

Successor(collection, x)the element of collection that follows x, that is $At(collection, k + 1)$ where x is the $k$-th element.

CombinatoricaModule[{c = collection, x = x, n, m, ind, refl}, n = Round[Log[2, Length[c]]]; m = Max[Append[Flatten[c], 0]]; ind[s_] := Table[Boole[MemberQ[s, i]], {i, n}]; refl[s_] := Sort[n + 1 - s]; Which[c === Permutations[Range[Length[x]]], NextPermutation[x], c === Partitions[Total[x]], NextPartition[x], c === Compositions[Total[x], Length[x]], NextComposition[x], c === Flatten[Map[Tableaux, Partitions[Length[Flatten[x]]]], 1], NextTableau[x], c === Subsets[Range[n]], NextSubset[Range[n], x], c === KSubsets[Range[m], Length[x]], NextKSubset[Range[m], x], c === Map[ind, BinarySubsets[Range[n]]], ind[NextBinarySubset[Range[n], Flatten[Position[x, 1]]]], c === Map[refl, GrayCodeSubsets[Range[n]]], NextGrayCodeSubset[Range[n], x], True, Missing["NotCombinatorica"]]]

Typed signatures

Type	Package	Applies
(indexed_collection<any>, any) -> unknown	combinatorics	always

Domain: Collections

Details

• The next element in the collection's order, by rank: $Successor(c, x) = At(c, k + 1)$ where $k$ is the position of $x$ in $c$. Families rank by arithmetic, so the call never builds the collection: the successor of the second-to-last permutation of twelve things is immediate.
• Past the last element it stays unevaluated, and so does an $x$ that is not an element of $c$. There is no wraparound: write it with At and Mod, as in $At(c, Mod(IndexOf(c, x), Count(c)) + 1)$.
• Defined in Epsil as At at the position IndexOf finds, one on; a declared family answers by its rank kernel, and tests/successor.test.ts holds the two together at every element of several small families.
• The order is the collection's own: lexicographic for KSubsets and SymmetricGroup, largest part first for IntegerPartitions, and each family's record says the rest. Combinatorica's nine Next* functions are this one head over its listing orders; they wrap at the end, and NextTableau stays inside one shape where StandardTableaux goes on to the next.
• Also reads a plain list: $Successor({5, 7, 9}, 7)$ is $9$.
• Not a Wolfram head: Wolfram has no successor of a collection element.

Examples

Basic #

the permutation after $1432$ in lexicographic order

live ↗

In$\mathrm{Successor}(\mathrm{SymmetricGroup}(4), {\operatorname{Permutation}}([1, 4, 3, 2]))$Out${\operatorname{Permutation}}([2, 1, 3, 4])$

partitions of 6 run largest part first, so $4 + 2$ is followed by $4 + 1 + 1$

live ↗

In$\mathrm{Successor}(\mathrm{IntegerPartitions}(6), {\operatorname{IntegerPartition}}([4, 2]))$Out${\operatorname{IntegerPartition}}([4, 1, 1])$

the first weak composition of 3 into 3 parts is followed by $0 + 1 + 2$

live ↗

In$\mathrm{Successor}(\mathrm{WeakCompositions}(3, 3), \bigl\lbrack0, 0, 3\bigr\rbrack)$Out$\bigl\lbrack0, 1, 2\bigr\rbrack$

within the shape $(4, 1)$

live ↗

In$\mathrm{Successor}(\mathrm{StandardTableaux}(5), {\operatorname{StandardTableau}}([[1, 3, 4, 5], [2]]))$Out${\operatorname{StandardTableau}}([[1, 2, 4, 5], [3]])$

subsets run by size, then lexicographically

live ↗

In$\mathrm{Successor}(\mathrm{Subsets}(3), \operatorname{Finset}(\operatorname{Tuple}(3,\bigl\lbrack1, 2\bigr\rbrack)))$Out$\operatorname{Finset}(\operatorname{Tuple}(3,\bigl\lbrack1, 3\bigr\rbrack))$

the 2-subsets of $\{1, \dots, 5\}$ in lexicographic order

live ↗

In$\mathrm{Successor}(\mathrm{KSubsets}(5, 2), \operatorname{Finset}(\operatorname{Tuple}(5,\bigl\lbrack2, 5\bigr\rbrack)))$Out$\operatorname{Finset}(\operatorname{Tuple}(5,\bigl\lbrack3, 4\bigr\rbrack))$

binary counting: $011$ is followed by $100$

live ↗

In$\mathrm{Successor}(\mathrm{BinaryStrings}(3), \bigl\lbrack0, 1, 1\bigr\rbrack)$Out$\bigl\lbrack1, 0, 0\bigr\rbrack$

consecutive subsets in Gray-code order differ by one element

live ↗

In$\mathrm{Successor}(\mathrm{GrayCodeSubsets}(3), \bigl\lbrack1, 2\bigr\rbrack)$Out$\bigl\lbrack2\bigr\rbrack$

Scope #

a list is ordered by its positions

live ↗

In$\mathrm{Successor}(\bigl\lbrack5, 7, 9\bigr\rbrack, 7)$Out$9$

found by rank, without building the 479001600 permutations of 12

live ↗

In$\mathrm{Successor}(\mathrm{SymmetricGroup}(12), \mathrm{SymmetricGroup}(12)[479\,001\,599])$Out${\operatorname{Permutation}}([12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1])$

Properties #

weak compositions are not in lexicographic order: $0 + 1 + 2$ is followed by $1 + 0 + 2$, not $0 + 2 + 1$

live ↗

In$\mathrm{Successor}(\mathrm{WeakCompositions}(3, 3), \bigl\lbrack0, 1, 2\bigr\rbrack)$Out$\bigl\lbrack1, 0, 2\bigr\rbrack$

the last tableau of one shape is followed by the first of the next

live ↗

In$\mathrm{Successor}(\mathrm{StandardTableaux}(4), {\operatorname{StandardTableau}}([[1, 2, 3], [4]]))$Out${\operatorname{StandardTableau}}([[1, 3], [2, 4]])$

the sixth subset in Gray-code order is followed by $\{1, 3\}$

live ↗

In$\mathrm{Successor}(\mathrm{GrayCodeSubsets}(3), \bigl\lbrack1, 2, 3\bigr\rbrack)$Out$\bigl\lbrack1, 3\bigr\rbrack$

the last element has no successor; the call is left as written

live ↗

In$\mathrm{Successor}(\mathrm{KSubsets}(5, 2), \operatorname{Finset}(\operatorname{Tuple}(5,\bigl\lbrack4, 5\bigr\rbrack)))$Out$\mathrm{Successor}(\mathrm{KSubsets}(5, 2), \operatorname{Finset}(\operatorname{Tuple}(5,\bigl\lbrack4, 5\bigr\rbrack)))$

a 3-subset is no element of the 2-subsets

live ↗

In$\mathrm{Successor}(\mathrm{KSubsets}(5, 2), \operatorname{Finset}(\operatorname{Tuple}(5,\bigl\lbrack1, 2, 3\bigr\rbrack)))$Out$\mathrm{Successor}(\mathrm{KSubsets}(5, 2), \operatorname{Finset}(\operatorname{Tuple}(5,\bigl\lbrack1, 2, 3\bigr\rbrack)))$

wrapping around is $At$ and $Mod$: after the last 2-subset comes the first

live ↗

In$\mathrm{KSubsets}(5, 2)[\mathrm{IndexOf}(\mathrm{KSubsets}(5, 2), \operatorname{Finset}(\operatorname{Tuple}(5,\bigl\lbrack4, 5\bigr\rbrack)))\bmod\mathrm{Count}(\mathrm{KSubsets}(5, 2))+1]$Out$\operatorname{Finset}(\operatorname{Tuple}(5,\bigl\lbrack1, 2\bigr\rbrack))$

About

Examples	16
Checked against	Combinatorica 9 agree4 differ2 unevaluated1 error
Crosswalk	Combinatorica ×10

Implementation

mappedcombinatoricapackages/oracle/src/mappings.ts

One head for Combinatorica's Next* functions. The template recognizes the collection by its Combinatorica listing and calls the matching function, which wraps around at the last element where ours stays unevaluated; the collection and element are otherwise read as the family bindings read them.

See also: At, IndexOf, Count, Mod, KSubsets, SymmetricGroup, IntegerPartitions