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Change EnumeratorOfCombinations(mset) to make it equal to Combinations(mset) #6797

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Change EnumeratorOfCombinations(mset) to make it equal to Combinations(mset) #6797

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9af26e3
Make `EnumeratorOfCombinations(mset)` equal to `Combinations(mset)`
fingolfin Oct 27, 2026
aa135c0
combinat: Clarify the EnumeratorOfCombinations docs
fingolfin Oct 30, 2026
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13 changes: 6 additions & 7 deletions lib/combinat.gd
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Expand Up @@ -341,11 +341,10 @@ DeclareGlobalFunction("Combinations");
## all combinations.
## <P/>
## <Ref Func="EnumeratorOfCombinations"/> returns an <Ref Attr="Enumerator" />
## for the same combinations.
## With second argument <A>k</A>, it is strictly sorted.
## <P/>
## The ordering of combinations from these functions can be different and also
## different from the list returned by <Ref Func="Combinations"/>.
## for the same combinations; the enumerator is equal to the strictly sorted
## list <C>Combinations( <A>mset</A>[, <A>k</A>] )</C>.
## <Ref Func="IteratorOfCombinations"/> produces them in a different order,
## without <A>k</A> by increasing size.
## <P/>
## <Example>
## gap> m:=[1..15];; Add(m, 15);
Expand All @@ -356,9 +355,9 @@ DeclareGlobalFunction("Combinations");
## 49152
## gap> cm := EnumeratorOfCombinations(m);;
## gap> cm[1000];
## [ 1, 2, 3, 6, 7, 8, 9, 10 ]
## [ 1, 2, 3, 4, 5, 7, 9, 10, 12, 13, 15, 15 ]
## gap> Position(cm, [1,13,15,15]);
## 36866
## 24572
## gap> cm := EnumeratorOfCombinations([1..30], 6);
## EnumeratorOfCombinations( [ 1 .. 30 ], 6 )
## gap> Length(cm);
Expand Down
212 changes: 156 additions & 56 deletions lib/combinat.gi
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Expand Up @@ -363,8 +363,123 @@ BindGlobal("EnumeratorOfCombinations_k", function(mset, k)
return enum;
end);

# All combinations are numbered in lexicographic order, in which a list
# comes before its extensions. Let els = Set(mset), l = Length(els), let
# max[i] be the multiplicity of els[i] in mset, and let
# S[i] = Product(max{[i..l]} + 1) be the number of sub-multisets of mset
# with entries in els{[i..l]}. A combination with d[i] copies of els[i],
# the last nonzero one d[t], comes after its Sum_i d[i] proper prefixes
# and, for each i < t, after the (max[i] - d[i]) * S[i+1] combinations
# that agree with it below els[i] and have more copies of els[i]. For a
# set of size n, S[i] = 2^(n+1-i).
BindGlobal("ElementNumber_AllCombinations_set", function(enum, num)
local set, s, r, comb, i;
if num > enum!.size then
Error("Index ", num, " not bound, expected at most ", enum!.size);
fi;
set := enum!.set;
s := enum!.size;
r := num-1;
comb := [];
i := 0;
while r > 0 do
i := i+1;
s := QuoInt(s, 2);
# r is the rank among the combinations of set{[i..n]}: [ ], the s
# with set[i], then the s-1 nonempty ones of set{[i+1..n]}
if r <= s then
Add(comb, set[i]);
r := r-1;
else
r := r-s;
fi;
od;
return comb;
end);

BindGlobal("NumberElement_AllCombinations_set", function(enum, comb)
local set, n, num, prev, x, p;
if not IsDenseList(comb) then
return fail;
fi;
set := enum!.set;
n := Length(set);
num := enum!.size + Length(comb) + 1;
prev := 0;
# with positions p_1 < .. < p_k, the sum of 2^(n-i) over the i < p_k
# other than the p_j is 2^n - 2^(n-p_k) - Sum_j 2^(n-p_j)
for x in comb do
p := Position(set, x);
if p = fail or p <= prev then
return fail;
fi;
num := num - 2^(n-p);
prev := p;
od;
return num - 2^(n-prev);
end);

BindGlobal("ElementNumber_AllCombinations_mset", function(enum, num)
local els, max, s, r, comb, i, b;
if num > enum!.size then
Error("Index ", num, " not bound, expected at most ", enum!.size);
fi;
els := enum!.els;
max := enum!.max;
s := enum!.size;
r := num-1;
comb := [];
i := 0;
while r > 0 do
i := i+1;
s := QuoInt(s, max[i]+1);
# r is the rank among the combinations of els{[i..l]}: els[i]^a for
# a = 0 .. max[i], then for a = max[i], .., 0 the s-1 combinations of
# els[i]^a followed by a nonempty one of els{[i+1..l]}
if r <= max[i] then
Append(comb, ListWithIdenticalEntries(r, els[i]));
r := 0;
else
b := QuoInt(r-max[i]-1, s-1);
Append(comb, ListWithIdenticalEntries(max[i]-b, els[i]));
r := r - max[i] - b*(s-1);
fi;
od;
return comb;
end);

BindGlobal("NumberElement_AllCombinations_mset", function(enum, comb)
local els, max, d, prev, x, p, num, i;
if not IsDenseList(comb) then
return fail;
fi;
els := enum!.els;
max := enum!.max;
d := 0*max;
prev := 1;
for x in comb do
if x <> els[prev] then
p := Position(els, x);
if p = fail or p < prev then
return fail;
fi;
prev := p;
fi;
if d[prev] = max[prev] then
return fail;
fi;
d[prev] := d[prev]+1;
od;
# Sum_{i<t} (max[i] - d[i]) * S[i+1] by Horner's rule, with t = prev
num := 0;
for i in [1..prev-1] do
num := num * (max[i]+1) + max[i] - d[i];
od;
return num * Product(max{[prev..Length(max)]} + 1) + Length(comb) + 1;
end);

InstallGlobalFunction(EnumeratorOfCombinations, function(mset, arg...)
local c, max, l, mods, size, els, ElementNumber, NumberElement;
local els, fam, max, enum;
if Length(arg) > 1 then
Error("usage: EnumeratorOfCombinations( <mset>[, <k>] )");
elif Length(arg) = 1 then
Expand All @@ -373,61 +488,46 @@ InstallGlobalFunction(EnumeratorOfCombinations, function(mset, arg...)
fi;
return EnumeratorOfCombinations_k(mset, arg[1]);
fi;
c := Collected(mset);
max := List(c, a-> a[2]);
els := List(c, a-> a[1]);
l := Length(max);
mods := max+1;
size := Product(mods);
# a combination can contain els[i] from 0 to max[i] times (mods[i]
# possibilities), we number the combination that contains a[i] times els[i]
# for all i by n = 1 + sum_i a[i]*m[i] where m[i] = prod_(j<i) mods[i]
ElementNumber := function(enu, n)
local comb, res, i, j;
if n > size then
Error("Index ", n, " not bound.");
fi;
comb := EmptyPlist(l);
n := n-1;
for i in [1..l] do
comb[i] := n mod mods[i];
n := (n - comb[i])/mods[i];
od;
res := [];
for i in [1..l] do
for j in [1..comb[i]] do
Add(res, els[i]);
od;
od;
return res;
end;
NumberElement := function(enu, comb)
local c, d, pos, n, a, i;
if not IsList(comb) then
return fail;
fi;
c := Collected(comb);
d := 0*max;
for a in c do
pos := PositionSorted(els, a[1]);
if not IsBound(els[pos]) or els[pos] <> a[1] or a[2] > max[pos] then
return fail;
else
d[pos] := a[2];
fi;
od;
n := 0;
for i in [l,l-1..1] do
n := n*mods[i] + d[i];
od;
return n+1;
end;
return EnumeratorByFunctions(ListsFamily, rec(
ElementNumber := ElementNumber,
NumberElement := NumberElement,
els := els,
Length := x->size,
max := max));
els := Set(mset);
# [ ] and nonempty lists lie in different families
if IsEmpty(els) then
fam := FamilyObj([ [] ]);
else
fam := ListsFamily;
fi;
if Length(els) = Length(mset) then
enum := EnumeratorByFunctions(fam, rec(
ElementNumber := ElementNumber_AllCombinations_set,
NumberElement := NumberElement_AllCombinations_set,
Length := enum -> enum!.size,
PrintObj := function(enum)
Print("EnumeratorOfCombinations( ");
PrintObj(enum!.set);
Print(" )");
end,
set := els,
size := 2^Length(els)));
else
max := List(Collected(mset), a -> a[2]);
enum := EnumeratorByFunctions(fam, rec(
ElementNumber := ElementNumber_AllCombinations_mset,
NumberElement := NumberElement_AllCombinations_mset,
Length := enum -> enum!.size,
PrintObj := function(enum)
Print("EnumeratorOfCombinations( ");
PrintObj(Concatenation(List([1..Length(enum!.els)],
i -> ListWithIdenticalEntries(enum!.max[i], enum!.els[i]))));
Print(" )");
end,
els := els,
max := max,
size := Product(max+1)));
fi;
SetIsSSortedList(enum, true);
if IsRange(els) or ForAll(els, IsSmallIntRep) then
SetFilterObj(enum, IsEnumeratorOfSmallIntLists);
fi;
return enum;
end);

BindGlobal("NextIterator_Combinations_set", function(it)
Expand Down
48 changes: 48 additions & 0 deletions tst/testinstall/combinat.tst
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Expand Up @@ -130,6 +130,54 @@ gap> for n in [ 0 .. 10 ] do
> Error( "different elements" );
> fi;
> od;
gap> for mset in Concatenation( [ [ 1 .. 10 ], [ 5, 3, 1 ], "abracadabra",
> [ (1,2), (), (1,2,3) ], [ (1,2), (), (1,2), (1,2,3) ] ],
> List( [ 0 .. 12 ],
> n -> List( [ 1 .. n ], i -> Random( [ 1 .. n ] ) ) ) ) do
> comb1:= Combinations( mset );
> enum:= EnumeratorOfCombinations( mset );
> if not ( HasIsSSortedList( enum ) and IsSSortedList( enum )
> and Length( enum ) = Length( comb1 )
> and AsList( enum ) = comb1 and enum = comb1
> and List( comb1, c -> Position( enum, c ) )
> = [ 1 .. Length( comb1 ) ] ) then
> Error( "wrong enumerator for ", mset );
> fi;
> od;
gap> enum:= EnumeratorOfCombinations( [ 2, 1, 1 ] );
EnumeratorOfCombinations( [ 1, 1, 2 ] )
gap> AsList( enum );
[ [ ], [ 1 ], [ 1, 1 ], [ 1, 1, 2 ], [ 1, 2 ], [ 2 ] ]
gap> IsEnumeratorOfSmallIntLists( enum );
true
gap> List( [ [ 2, 1 ], [ 1, 1, 1 ], [ 3 ], [ 1, 1, 2, 2 ], 1 ],
> c -> Position( enum, c ) );
[ fail, fail, fail, fail, fail ]
gap> enum[ 7 ];
Error, Index 7 not bound, expected at most 6
gap> EnumeratorOfCombinations( [ 1 .. 3 ] )[ 9 ];
Error, Index 9 not bound, expected at most 8
gap> enum:= EnumeratorOfCombinations( [ 1 .. 100 ] );
EnumeratorOfCombinations( [ 1 .. 100 ] )
gap> enum[ 2^99 + 1 ];
[ 1, 100 ]
gap> enum[ 2^100 ];
[ 100 ]
gap> Position( enum, [ 2 ] ) = 2^99 + 2;
true
gap> List( [ [ 2, 1 ], [ 1, 1 ], [ 101 ], [ 1 .. 101 ], 1 ],
> c -> Position( enum, c ) );
[ fail, fail, fail, fail, fail ]
gap> mset:= Concatenation( [ 1 .. 50 ], [ 1 .. 50 ] );;
gap> enum:= EnumeratorOfCombinations( mset );;
gap> Length( enum ) = 3^50;
true
gap> enum[ 3^50 ];
[ 50, 50 ]
gap> Position( enum, [ 1, 2, 2, 50 ] ) = 3^49 + 3^48 + 2;
true
gap> enum[ 3^49 + 3^48 + 2 ];
[ 1, 2, 2, 50 ]
gap> for mset in [ [], [ 1 .. 7 ], [ 5, 3, 1 ], [ 1, 1, 2, 3, 3, 3, 4 ],
> "abracadabra", [ (1,2), (), (1,2), (1,2,3) ] ] do
> for k in [ 0 .. Length( mset ) + 1 ] do
Expand Down
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