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combinationsthat

Combinationsthat is a label that may appear in informal mathematics and computer science to refer to the set of all k-element combinations that meet a specified condition. It is not a widely standardized term, but it is used to describe a predicate-driven approach to selecting subsets from a finite ground set.

Formal definition: Let S be a finite set with n elements, k an integer with 0 ≤ k

Examples: If S = {1, 2, 3, 4}, k = 2, and P(T) asserts that the sum of the

Computation and applications: Enumerating combinationsthat is a common subproblem in combinatorial optimization and constraint satisfaction. Algorithms

See also: combinatorics, k-combinations, constrained subset selection.

≤
n,
and
P
a
predicate
defined
on
k-element
subsets
of
S.
The
collection
of
interest,
denoted
as
combinationsthat(S,
k,
P),
consists
of
all
T
⊆
S
with
|T|
=
k
and
P(T)
is
true.
This
view
treats
combinationsthat
as
a
constrained
subset
of
the
k-combinations
of
S.
two
elements
is
even,
then
combinationsthat(S,
2,
P)
=
{{1,
3},
{2,
4}}.
Another
example:
S
as
a
set
of
tasks
with
durations,
P(T)
holds
when
the
total
duration
of
the
k
tasks
is
within
a
target
window.
include
backtracking,
pruning
based
on
P,
and
generation
with
test,
or
specialized
techniques
like
meet-in-the-middle
for
large
n.
Applications
span
scheduling,
experimental
design,
test
case
generation,
and
resource
allocation.