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JNumberTools - Validation Rules

All operations throw NullPointerException on null input (programmer error).
Empty collections/maps are allowed and follow the rules below.
Negative counts/ranges/frequencies throw IllegalArgumentException.

UNIQUE COMBINATION (ⁿCᵣ)

n (set size) r (selection) Mathematical Count Iterator Returns Interpretation
n = 0 r = 0 ⁰C₀ = 1 1 [[]] One empty combination
n = 0 r > 0 ⁰Cᵣ = 0 0 [] Cannot select from empty set
n > 0 r = 0 ⁿC₀ = 1 1 [[]] One empty combination
n > 0 0 < r ≤ n ⁿCᵣ ⁿCᵣ combinations Normal case
n > 0 r > n ⁿCᵣ = 0 0 [] Selection exceeds set size
Exception Condition Throws
n < 0 IllegalArgumentException n must be ≥ 0
r < 0 IllegalArgumentException r must be ≥ 0
null input NullPointerException Elements list cannot be null

REPETITIVE COMBINATION (ⁿ⁺ʳ⁻¹Cᵣ)

n (set size) r (selection) Mathematical Count Iterator Returns Interpretation
n = 0 r = 0 (by convention) 1 [[]] One empty combination
n = 0 r > 0 0 0 [] Cannot select from empty set
n > 0 r = 0 1 1 [[]] One empty combination
n > 0 r > 0 ⁿ⁺ʳ⁻¹Cᵣ ⁿ⁺ʳ⁻¹Cᵣ combinations Normal case
Exception Condition Throws
n < 0 IllegalArgumentException n must be ≥ 0
r < 0 IllegalArgumentException r must be ≥ 0
null input NullPointerException Elements list cannot be null

UNIQUE PERMUTATION (n!)

n (set size) Mathematical Count Iterator Returns Interpretation
n = 0 0! = 1 1 [[]] One empty permutation
n > 0 n! n! permutations Normal case
Exception Condition Throws
n < 0 IllegalArgumentException
null input NullPointerException

Note: Input elements must be distinct; duplicates cause undefined behavior or IllegalArgumentException.

K-PERMUTATION (ⁿPₖ)

n (set size) k (selection) Mathematical Count Iterator Returns Interpretation
n = 0 k = 0 ⁰P₀ = 1 1 [[]] One empty permutation
n = 0 k > 0 ⁰Pₖ = 0 0 [] Cannot select from empty set
n > 0 k = 0 ⁿP₀ = 1 1 [[]] One empty permutation
n > 0 0 < k ≤ n ⁿPₖ ⁿPₖ permutations Normal case
n > 0 k > n ⁿPₖ = 0 0 [] Selection exceeds set size
Exception Condition Throws
n < 0 IllegalArgumentException n must be ≥ 0
k < 0 IllegalArgumentException k must be ≥ 0
null input NullPointerException Elements list cannot be null

REPETITIVE PERMUTATION (nʳ)

n (set size) r (length) Mathematical Count Iterator Returns Interpretation
n = 0 r = 0 0⁰ = 1 1 [[]] One empty permutation
n = 0 r > 0 0ʳ = 0 0 [] Cannot form from empty set
n > 0 r = 0 n⁰ = 1 1 [[]] One empty permutation
n > 0 r > 0 nʳ nʳ permutations Normal case
Exception Condition Throws
n < 0 IllegalArgumentException n must be ≥ 0
r < 0 IllegalArgumentException r must be ≥ 0
null input NullPointerException Elements list cannot be null

MULTISET PERMUTATION

Map State Mathematical Count Iterator Returns Interpretation
Empty map 0! = 1 1 [[]] One empty permutation
Non-empty n! / (Π fᵢ!) multinomial permutations Normal case
All frequencies = 0 treated as empty 1 [[]] Treated as empty map
Exception Condition Throws
any frequency < 0 IllegalArgumentException
null input NullPointerException

Note: Frequencies must be non-negative integers; zero means element not available.

MULTISET COMBINATION

Map State r (selection) Mathematical Count Iterator Returns Interpretation
Empty map r = 0 1 1 [{}] One empty multiset
Empty map r > 0 0 0 [] Cannot select from empty multiset
Non-empty r = 0 1 1 [{}] One empty multiset
Non-empty r > 0 multisetCombinationsCount(r, freq) calculated combinations Normal case
Non-empty r > ∑fᵢ 0 0 [] Selection exceeds available elements
Exception Condition Throws
any frequency < 0 IllegalArgumentException Frequencies must be ≥ 0
r < 0 IllegalArgumentException r must be ≥ 0
null input NullPointerException Map cannot be null

SUBSETS (Power Set)

n (set size) Range [from, to] Mathematical Count Iterator Returns Interpretation
n = 0 [0, 0] 2⁰ = 1 1 [[]] One empty subset
n = 0 [0, m] where m>0 Σ2⁰ = 1 1 [[]] Only empty subset exists
n = 0 [1, m] where m>0 0 0 [] No non-empty subsets
n > 0 [0, 0] 1 1 [[]] One empty subset
n > 0 [0, n] 2ⁿ 2ⁿ all subsets Full power set
n > 0 [a, b] where 0 ≤ a ≤ b ≤ n Σ ⁿCᵢ for i=a..b Σ ⁿCᵢ subsets in range Subsets of specific sizes
Exception Condition Throws
from < 0 IllegalArgumentException from must be ≥ 0
to < from IllegalArgumentException to must be ≥ from
from > n IllegalArgumentException from cannot exceed n
to > n IllegalArgumentException to cannot exceed n
null input NullPointerException Elements list cannot be null

SIMPLE CARTESIAN PRODUCT

Condition Count Iterator Returns Interpretation
Any dimension is empty (positive multiplicity) 0 [] Empty product
All dimensions contribute 1 tuple 1 [[]] Single empty tuple
Normal case (no empty dimensions) Π sizes products Normal product
Exception Condition Throws
null input for any dimension NullPointerException

CONSTRAINED CARTESIAN PRODUCT

Condition Count Iterator Returns Interpretation
Any dimension has count = 0 0 [] Empty product (impossible)
All dimensions have count = 1 1 [[]] Single empty tuple
Normal case Π counts products Normal constrained product
Exception Condition Throws
quantity < 0 (for distinct/multi-select) IllegalArgumentException
invalid range (from < 0, to < from, from > n, to > n) IllegalArgumentException
null input for any dimension NullPointerException

Note: Multiplicity 0 in a dimension (min = max = 0) contributes 1 empty tuple (as if dimension is absent).

GENERAL RULES

  1. Null input (collections, maps) always throws NullPointerException (programmer error).
  2. Empty list (∅) is allowed and follows the rules above.
  3. Negative counts, ranges, frequencies, or invalid parameters throw IllegalArgumentException.
  4. Zero frequency in multiset means element is not available (treated as absent).
  5. For zero selection (r=0, k=0, quantity=0): always count = 1, returns single empty result ([[]], [{}]).
  6. For positive selection with empty input: always count = 0, empty iterator ([]).