From 076b9af0c72a55d3cf861b365e241e33756a2ceb Mon Sep 17 00:00:00 2001
From: depial <91621102+depial@users.noreply.github.com>
Date: Tue, 23 Dec 2025 10:46:53 -0500
Subject: [PATCH 01/11] Add nthcombo functions

---
 src/combinations.jl | 97 +++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 97 insertions(+)

diff --git a/src/combinations.jl b/src/combinations.jl
index ec466a9..01c9d1b 100644
--- a/src/combinations.jl
+++ b/src/combinations.jl
@@ -273,3 +273,100 @@ function powerset(a, min::Integer=0, max::Integer=length(a))
     min < 1 && append!(itrs, eltype(a)[])
     Iterators.flatten(itrs)
 end
+
+
+# Nth Combination
+
+"""
+    nthcombo(a, k::Int, n::Int)
+
+Compute the `n`th lexicographic k-combination of the vector `a`.
+
+# Examples
+```jldoctest
+julia> collect(combinations([1,2,3], 2))
+3-element Vector{Vector{Int64}}:
+ [1, 2]
+ [1, 3]
+ [2, 3]
+
+julia> nthperm([1, 2, 3], 2, 1)
+2-element Vector{Int64}:
+ 1
+ 2
+
+julia> nthperm([1, 2, 3], 2, 2)
+2-element Vector{Int64}:
+ 1
+ 3
+
+julia> nthperm([1, 2, 3], 2, 4)
+ERROR: ArgumentError: combination k must satisfy 0 ≤ k ≤ 3, got 4
+[...]
+```
+"""
+function nthcombo(a, k::Int, n::Int)
+    len = length(a)
+    (k == 0 || k == len) && return collect(a)[1:k]
+    0 < k < len || throw(ArgumentError("combination k must satisfy 0 ≤ k ≤ $len, got $k"))
+    
+    combo = eltype(a)[]
+    sizehint!(combo, k)
+    ncombos = binomial(len-1, k-1)
+    for i in eachindex(a)
+        if n ≤ ncombos
+            @inbounds push!(combo, a[i])
+            isone(k) && return combo
+            k -= 1
+            ncombos *= k
+        else
+            n -= ncombos
+            ncombos *= len - k
+        end
+        len -= 1
+        ncombos ÷= len
+    end
+end
+
+"""
+    nthcombo(a, c::Vector)
+
+Return the integer `n` that generated index-based lexicographic combination `c` from `a`. 
+Note that `nthcombo(a, nthcombo(a, k, n)) == n` for `1 ≤ n ≤ binomial(length(a), k)` and `unique(a) == a`.
+In the case `unique(a) ≠ a`, returns the lowest `n` matching the combination, and 
+thus is not guaranteed to be the inverse of `nthcombo(a, k, n)`.
+
+# Examples
+```jldoctest
+julia> nthcombo([1:3...], nthcombo([1:3...], 2, 3))
+ 3
+
+julia> collect(combinations([1, 2, 3], 2))
+3-element Vector{Vector{Int64}}:
+ [1, 2]
+ [1, 3]
+ [2, 3]
+
+julia> nthcombo([1, 2, 3], [1, 2])
+ 1
+
+julia> nthcombo([1, 2, 3], [2, 3])
+ 3
+```
+"""
+function nthcombo(a, combo::Vector)
+    aunique = unique(a)
+    idxmap = Dict(zip(aunique, 1:length(aunique)))
+    idxs = [idxmap[v] for v in combo]
+    ranges = collect(zip([0; idxs[1:end-1]] .+ 1, idxs .- 1))
+    m, k = length(a), length(combo)
+    
+    n = 1
+    for i in 1:k
+        lower, upper = ranges[i]
+        if upper - lower ≥ 0
+            n += sum(binomial.(m .- lower:upper, k - i))
+        end
+    end
+    n
+end

From b9306c01f675e000a592ed3e4231e16daedccdc6 Mon Sep 17 00:00:00 2001
From: depial <91621102+depial@users.noreply.github.com>
Date: Tue, 23 Dec 2025 11:06:43 -0500
Subject: [PATCH 02/11] Update combinations.jl

Updates docstrings and exports nthcombo
---
 src/combinations.jl | 9 +++++----
 1 file changed, 5 insertions(+), 4 deletions(-)

diff --git a/src/combinations.jl b/src/combinations.jl
index 01c9d1b..daef7e8 100644
--- a/src/combinations.jl
+++ b/src/combinations.jl
@@ -2,7 +2,8 @@ export combinations,
        CoolLexCombinations,
        multiset_combinations,
        with_replacement_combinations,
-       powerset
+       powerset,
+       nthcombo
 
 #The Combinations iterator
 struct Combinations
@@ -290,17 +291,17 @@ julia> collect(combinations([1,2,3], 2))
  [1, 3]
  [2, 3]
 
-julia> nthperm([1, 2, 3], 2, 1)
+julia> nthcombo([1, 2, 3], 2, 1)
 2-element Vector{Int64}:
  1
  2
 
-julia> nthperm([1, 2, 3], 2, 2)
+julia> nthcombo([1, 2, 3], 2, 2)
 2-element Vector{Int64}:
  1
  3
 
-julia> nthperm([1, 2, 3], 2, 4)
+julia> nthcombo([1, 2, 3], 2, 4)
 ERROR: ArgumentError: combination k must satisfy 0 ≤ k ≤ 3, got 4
 [...]
 ```

From 0639a9868de53a53f3076b1477a7b43563c6ac4e Mon Sep 17 00:00:00 2001
From: depial <91621102+depial@users.noreply.github.com>
Date: Tue, 23 Dec 2025 11:31:25 -0500
Subject: [PATCH 03/11] Update nthcombo

Added `collect` for proper broadcasting behavior.
---
 src/combinations.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/combinations.jl b/src/combinations.jl
index daef7e8..59eaa17 100644
--- a/src/combinations.jl
+++ b/src/combinations.jl
@@ -366,7 +366,7 @@ function nthcombo(a, combo::Vector)
     for i in 1:k
         lower, upper = ranges[i]
         if upper - lower ≥ 0
-            n += sum(binomial.(m .- lower:upper, k - i))
+            n += sum(binomial.(m .- collect(lower:upper), k - i))
         end
     end
     n

From c560e5030022bcb26d7718657e9610402b9f17a1 Mon Sep 17 00:00:00 2001
From: depial <91621102+depial@users.noreply.github.com>
Date: Tue, 23 Dec 2025 11:42:11 -0500
Subject: [PATCH 04/11] Update combinations.jl

---
 src/combinations.jl | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/src/combinations.jl b/src/combinations.jl
index 59eaa17..7f45416 100644
--- a/src/combinations.jl
+++ b/src/combinations.jl
@@ -301,7 +301,7 @@ julia> nthcombo([1, 2, 3], 2, 2)
  1
  3
 
-julia> nthcombo([1, 2, 3], 2, 4)
+julia> nthcombo([1, 2, 3], 4, 2)
 ERROR: ArgumentError: combination k must satisfy 0 ≤ k ≤ 3, got 4
 [...]
 ```
@@ -340,7 +340,7 @@ thus is not guaranteed to be the inverse of `nthcombo(a, k, n)`.
 # Examples
 ```jldoctest
 julia> nthcombo([1:3...], nthcombo([1:3...], 2, 3))
- 3
+3
 
 julia> collect(combinations([1, 2, 3], 2))
 3-element Vector{Vector{Int64}}:
@@ -349,10 +349,10 @@ julia> collect(combinations([1, 2, 3], 2))
  [2, 3]
 
 julia> nthcombo([1, 2, 3], [1, 2])
- 1
+1
 
 julia> nthcombo([1, 2, 3], [2, 3])
- 3
+3
 ```
 """
 function nthcombo(a, combo::Vector)

From dd2c6ff3f6f7a788880c0fcfb4781555ef403995 Mon Sep 17 00:00:00 2001
From: depial <91621102+depial@users.noreply.github.com>
Date: Tue, 23 Dec 2025 13:20:13 -0500
Subject: [PATCH 05/11] add input validation

---
 src/combinations.jl | 8 ++++++--
 1 file changed, 6 insertions(+), 2 deletions(-)

diff --git a/src/combinations.jl b/src/combinations.jl
index 7f45416..90bb854 100644
--- a/src/combinations.jl
+++ b/src/combinations.jl
@@ -308,12 +308,15 @@ ERROR: ArgumentError: combination k must satisfy 0 ≤ k ≤ 3, got 4
 """
 function nthcombo(a, k::Int, n::Int)
     len = length(a)
+    0 ≤ k ≤ len || throw(ArgumentError("combination k must satisfy 0 ≤ k ≤ $len, got $k"))
+    ncombos = binomial(len, k)
+    0 < n ≤ ncombos || throw(ArgumentError("n must satisfy 0 < n ≤ $ncombos, got $n"))
     (k == 0 || k == len) && return collect(a)[1:k]
-    0 < k < len || throw(ArgumentError("combination k must satisfy 0 ≤ k ≤ $len, got $k"))
     
     combo = eltype(a)[]
     sizehint!(combo, k)
-    ncombos = binomial(len-1, k-1)
+    ncombos *= k
+    ncombos ÷= len
     for i in eachindex(a)
         if n ≤ ncombos
             @inbounds push!(combo, a[i])
@@ -356,6 +359,7 @@ julia> nthcombo([1, 2, 3], [2, 3])
 ```
 """
 function nthcombo(a, combo::Vector)
+    length(combo) ≤ length(a) && all(∈(a), combo) || throw(ArgumentError("$combo not a combination of $a"))
     aunique = unique(a)
     idxmap = Dict(zip(aunique, 1:length(aunique)))
     idxs = [idxmap[v] for v in combo]

From 2cd0a58d6286dba4ac6c127468bc0e573e279282 Mon Sep 17 00:00:00 2001
From: depial <91621102+depial@users.noreply.github.com>
Date: Tue, 23 Dec 2025 13:21:38 -0500
Subject: [PATCH 06/11] Add testing for nthcombo

---
 test/combinations.jl | 20 ++++++++++++++++++++
 1 file changed, 20 insertions(+)

diff --git a/test/combinations.jl b/test/combinations.jl
index f70a566..4ef520c 100644
--- a/test/combinations.jl
+++ b/test/combinations.jl
@@ -44,4 +44,24 @@
     @test collect(powerset(['a', 'b', 'c'], 1)) == Any[['a'], ['b'], ['c'], ['a', 'b'], ['a', 'c'], ['b', 'c'], ['a', 'b', 'c']]
     @test collect(powerset(['a', 'b', 'c'], 1, 2)) == Any[['a'], ['b'], ['c'], ['a', 'b'], ['a', 'c'], ['b', 'c']]
 
+    # Nth Combo
+    @test nthcombo([1, 2, 3, 4], 0, 1) == []
+    @test nthcombo([1, 2, 3, 4], 4, 1) == [1, 2, 3, 4]
+    @test nthcombo([1, 2, 3, 4], 3, 2) == [1, 2, 4]
+    @test all([nthcombo([1, 2, 3, 4], 2, n) for n in 1:binomial(4, 2)] .== collect(combinations([1, 2, 3, 4], 2)))
+    @test_throws ArgumentError nthcombo([1, 2, 3, 4], 0, 3)
+    @test_throws ArgumentError nthcombo([1, 2, 3, 4], 5, 3)
+    @test_throws ArgumentError nthcombo([1, 2, 3, 4], 2, 0)
+    @test_throws ArgumentError nthcombo([1, 2, 3], 2, 6)
+    
+    @test nthcombo([1, 2, 3, 4], []) == 1
+    @test nthcombo([1, 2, 3, 4], [1, 2, 3, 4]) == 1
+    @test nthcombo([1, 2, 3, 4], [1, 2, 4]) == 2
+    @test [nthcombo(1:7, combo) for combo in combinations(1:7, 3)] == collect(1:binomial(7, 3))
+    @test_throws ArgumentError nthcombo([1, 2, 3, 4], [1, 5])
+    @test_throws ArgumentError nthcombo([1, 2, 3], [1, 2, 3, 3])
+
+    data = collect(1:n)
+    @test all([nthcombo(data, nthcombo(data, k, j)) == j for k in 1:7 for j in 1:binomial(7, k)])
+    
 end

From ddcc94b76c9fff1e31737fa14cf1ab67fbf5edf1 Mon Sep 17 00:00:00 2001
From: depial <91621102+depial@users.noreply.github.com>
Date: Tue, 23 Dec 2025 13:24:35 -0500
Subject: [PATCH 07/11] Update combinations.jl

---
 test/combinations.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/test/combinations.jl b/test/combinations.jl
index 4ef520c..2e230e3 100644
--- a/test/combinations.jl
+++ b/test/combinations.jl
@@ -61,7 +61,7 @@
     @test_throws ArgumentError nthcombo([1, 2, 3, 4], [1, 5])
     @test_throws ArgumentError nthcombo([1, 2, 3], [1, 2, 3, 3])
 
-    data = collect(1:n)
+    data = collect(1:7)
     @test all([nthcombo(data, nthcombo(data, k, j)) == j for k in 1:7 for j in 1:binomial(7, k)])
     
 end

From bed7ccdaa7573666446ac9c49dd304d703a3939b Mon Sep 17 00:00:00 2001
From: depial <91621102+depial@users.noreply.github.com>
Date: Tue, 23 Dec 2025 13:37:48 -0500
Subject: [PATCH 08/11] Update combinations.jl

---
 src/combinations.jl | 1 +
 1 file changed, 1 insertion(+)

diff --git a/src/combinations.jl b/src/combinations.jl
index 90bb854..953c8cc 100644
--- a/src/combinations.jl
+++ b/src/combinations.jl
@@ -359,6 +359,7 @@ julia> nthcombo([1, 2, 3], [2, 3])
 ```
 """
 function nthcombo(a, combo::Vector)
+    isempty(combo) && return 1   
     length(combo) ≤ length(a) && all(∈(a), combo) || throw(ArgumentError("$combo not a combination of $a"))
     aunique = unique(a)
     idxmap = Dict(zip(aunique, 1:length(aunique)))

From 78e73aef18cd4beffe359a4ec70e777506cf2283 Mon Sep 17 00:00:00 2001
From: depial <91621102+depial@users.noreply.github.com>
Date: Tue, 23 Dec 2025 15:31:22 -0500
Subject: [PATCH 09/11] add helper function iscombo

---
 src/combinations.jl | 11 +++++++++--
 1 file changed, 9 insertions(+), 2 deletions(-)

diff --git a/src/combinations.jl b/src/combinations.jl
index 953c8cc..1ad6bd0 100644
--- a/src/combinations.jl
+++ b/src/combinations.jl
@@ -359,8 +359,8 @@ julia> nthcombo([1, 2, 3], [2, 3])
 ```
 """
 function nthcombo(a, combo::Vector)
-    isempty(combo) && return 1   
-    length(combo) ≤ length(a) && all(∈(a), combo) || throw(ArgumentError("$combo not a combination of $a"))
+    iscombo(a, combo) || throw(ArgumentError("$combo not a combination of $a"))
+
     aunique = unique(a)
     idxmap = Dict(zip(aunique, 1:length(aunique)))
     idxs = [idxmap[v] for v in combo]
@@ -376,3 +376,10 @@ function nthcombo(a, combo::Vector)
     end
     n
 end
+
+function iscombo(a, combo)
+    counts = Dict{eltype(a), Int}()
+    foreach(key -> counts[key] = get(counts, key, 0) + 1, a)
+    foreach(key -> counts[key] = get(counts, key, 0) - 1, combo)
+    all(≥(0), (0, values(counts)...))
+end

From 085116414e6b33c6f58f1674527f1aa72bbad241 Mon Sep 17 00:00:00 2001
From: depial <91621102+depial@users.noreply.github.com>
Date: Tue, 23 Dec 2025 15:35:05 -0500
Subject: [PATCH 10/11] adjust anonymous function for Julia 1.0

---
 src/combinations.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/combinations.jl b/src/combinations.jl
index 1ad6bd0..2c02bc5 100644
--- a/src/combinations.jl
+++ b/src/combinations.jl
@@ -381,5 +381,5 @@ function iscombo(a, combo)
     counts = Dict{eltype(a), Int}()
     foreach(key -> counts[key] = get(counts, key, 0) + 1, a)
     foreach(key -> counts[key] = get(counts, key, 0) - 1, combo)
-    all(≥(0), (0, values(counts)...))
+    all(v -> v ≥ 0, (0, values(counts)...))
 end

From f6c8eb2ab8951f8c7c423ba1f169ef8cc2b4275f Mon Sep 17 00:00:00 2001
From: depial <91621102+depial@users.noreply.github.com>
Date: Tue, 23 Dec 2025 15:37:46 -0500
Subject: [PATCH 11/11] support for Julia 1.0

---
 src/combinations.jl | 1 +
 1 file changed, 1 insertion(+)

diff --git a/src/combinations.jl b/src/combinations.jl
index 2c02bc5..9237f41 100644
--- a/src/combinations.jl
+++ b/src/combinations.jl
@@ -359,6 +359,7 @@ julia> nthcombo([1, 2, 3], [2, 3])
 ```
 """
 function nthcombo(a, combo::Vector)
+    isempty(combo) && return 1
     iscombo(a, combo) || throw(ArgumentError("$combo not a combination of $a"))
 
     aunique = unique(a)