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numbergives

Numbergives is a term used in combinatorics and theoretical computer science to describe a class of problems in which a target integer is distributed among a collection of recipients under specified constraints. The concept treats the allocation as a kind of “gift” that must sum to the target, and it is studied as a generalization of number partitions and compositions.

Definition and variants: An n-numbergive consists of a multiset of nonnegative integers {a1, a2, ..., ak} such

Examples: For n = 4 and two parts with nonnegativity and no further constraints, the ordered numbergives

History and usage: The term numbergives emerged in online mathematical communities in the early 2020s as a

Properties and computation: Counting numbergives often employs generating functions or recurrence relations. Depending on the constraints,

See also: partitions, compositions, integer partition problem, generating functions, dynamic programming.

that
the
sum
of
the
parts
equals
n,
together
with
a
set
of
constraints
C.
When
the
order
of
the
parts
matters,
the
problem
corresponds
to
compositions
of
n
with
constraints;
when
the
order
is
ignored,
it
aligns
with
partitions
with
constraints.
Common
constraints
include
nonnegativity,
distinctness
of
parts,
prescribed
parity,
or
bounds
on
the
number
of
parts.
are
(0,4),
(1,3),
(2,2),
(3,1),
(4,0).
If
order
is
ignored,
the
distinct
numbergives
are
{0,4},
{1,3},
and
{2,2}.
Imposing
distinctness
would
remove
(2,2),
leaving
{0,4}
and
{1,3}.
flexible
label
for
partition-
and
composition-like
problems
with
extra
constraints.
It
is
used
in
instructional
contexts
to
illustrate
counting
methods,
generating
functions,
and
dynamic
programming
approaches.
the
counting
problem
can
range
from
straightforward
to
computationally
challenging,
with
some
cases
falling
into
classic
partition
or
composition
problems
and
others
requiring
specialized
algorithms.