subsequence. 
Subsequences
Subsequences[list]
gives the list of all possible subsequences of list.
Subsequences[list,n]
gives all subsequences containing at most n elements.
Subsequences[list,{n}]
gives all subsequences containing exactly n elements.
Subsequences[list,{nmin,nmax}]
gives all subsequences containing between nmin and nmax elements.
Subsequences[list,nspec,s]
limits the result to the first s subsequences.
Subsequences[list,nspec,{s}]
gives if possible the s 
subsequence.
Details
- Subsequences[list,All] is equivalent to Subsequences[list].
- Subsequences[list,{nmin,nmax,dn}] gives subsequences containing nmin, nmin+dn, … elements.
- Subsequences[list,nspec,All] is equivalent to Subsequences[list,nspec].
- Subsequences[list,nspec,spec] gives the same result as Take[Subsequences[list,nspec],spec], provided that the elements specified by spec are present.
- Subsequences[list,nspec,UpTo[s]] returns s subsequences, or as many as are available.
- The head of list in Subsequences[list,nspec,spec] does not need to be List.
- Subsequences[BioSequence["type","seq"],nspec,…] gives the specified subsequences of the given BioSequence.
Examples
open all close allBasic Examples (3)
Subsequences[{a, b, c}]All possible subsequences containing up to 2 elements:
Subsequences[{a, b, c, d}, 2]Subsequences containing exactly 2 elements:
Subsequences[{a, b, c, d}, {2}]All subsequences of a biomolecular sequence containing exactly three elements:
Subsequences[BioSequence["DNA", "CGGTGA"], {3}]Scope (6)
All subsequences of {a,b,c,d}:
Subsequences[{a, b, c, d}]The first 2 subsequences containing 3 elements:
Subsequences[{a, b, c, d, e}, {3}, 2]All subsequences with even length:
Subsequences[{a, b, c, d, e}, {0, 5, 2}]
subsequence:Subsequences[Range[20], All, {181}]The odd-numbered subsequences of {a,b,c,d} in reverse order:
Subsequences[{a, b, c, d}, All, {11, 1, -2}]Use Subsequences with UpTo:
Subsequences[{a, b, c, d}, All, UpTo[20]]Generalizations & Extensions (1)
Applications (2)
Use Subsequences to obtain all subsequences common to two lists:
commonSubsequences[list1_, list2_] := Intersection[Subsequences[list1], Subsequences[list2]];
commonSubsequences[{a, b, c, e, d}, {b, c, e, a, d}]Or specify the length of the common subsequences to consider:
commonSubsequences[list1_, list2_, n_] := Intersection[Subsequences[list1, {n}], Subsequences[list2, {n}]];
commonSubsequences[{a, b, c, e, d}, {b, c, e, a, d}, 2]Compare to LongestCommonSubsequence:
LongestCommonSubsequence[{a, b, c, e, d}, {b, c, e, a, d}]Construct the boundary of a hexagon and color its sides randomly:
Graphics[{RandomColor[], Line[#]}& /@ Subsequences[Table[{Cos[2Pi i / 6], Sin[2 Pi i / 6]}, {i, 7}], {2}]]Properties & Relations (5)
Subsequences is equivalent to a form of Partition:
Subsequences[{a, b, c, d}, {2}]Partition[{a, b, c, d}, 2, 1]Subsequences preserves the order of the input:
Subsequences[{c, b, a}]Different occurrences of the same element are treated as distinct:
Subsequences[{a, b, b, b}]SequenceCases can also find the subsequences of a list:
SequenceCases[{a, b, c}, _, Overlaps -> All]Subsequences[{a, b, c}]Construct a 3x3 Hilbert matrix:
Subsequences[Table[1 / n, {n, 1, 5}], {3}]//MatrixFormHilbertMatrix[3]//MatrixFormPossible Issues (2)
Subsequences[list,nspec,spec] only evaluates when all requested items are present:
Subsequences[Range[3], All, 10]
Subsequences generates only one list of length 0:
Subsequences[{1, 1, 2}]This follows the behavior of Subsets:
Subsets[{1, 1, 2}]Text
Wolfram Research (2016), Subsequences, Wolfram Language function, https://reference.wolfram.com/language/ref/Subsequences.html (updated 2020).
CMS
Wolfram Language. 2016. "Subsequences." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/Subsequences.html.
APA
Wolfram Language. (2016). Subsequences. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Subsequences.html