er

Author: PeterLuschny

Manifest.toml

@@ -19,12 +19,6 @@ version = "0.4.3"
 deps = ["Compat", "CredentialsHandler", "Libdl", "Pkg", "SHA", "TOML", "Test"]
 uuid = "b99e7846-7c00-51b0-8f62-c81ae34c0232"
 
-[[Combinatorics]]
-deps = ["LinearAlgebra", "Polynomials", "Test"]
-git-tree-sha1 = "50b3ae4d643dc27eaff69fb6be06ee094d5500c9"
-uuid = "861a8166-3701-5b0c-9a16-15d98fcdc6aa"
-version = "0.7.0"
-
 [[Compat]]
 deps = ["Base64", "Dates", "DelimitedFiles", "Distributed", "InteractiveUtils", "LibGit2", "Libdl", "LinearAlgebra", "Markdown", "Mmap", "Pkg", "Printf", "REPL", "Random", "Serialization", "SharedArrays", "Sockets", "SparseArrays", "Statistics", "Test", "UUIDs", "Unicode"]
 git-tree-sha1 = "84aa74986c5b9b898b0d1acaf3258741ee64754f"
@@ -121,12 +115,6 @@ version = "0.3.7"
 deps = ["BinaryProvider", "CredentialsHandler", "Dates", "HTTP", "LibGit2", "Markdown", "Printf", "REPL", "Random", "SHA", "TOML", "UUIDs"]
 uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"
 
-[[Polynomials]]
-deps = ["LinearAlgebra", "SparseArrays", "Test"]
-git-tree-sha1 = "62142bd65d3f8aeb2226ec64dd8493349147df94"
-uuid = "f27b6e38-b328-58d1-80ce-0feddd5e7a45"
-version = "0.5.2"
-
 [[Printf]]
 deps = ["Unicode"]
 uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

Project.toml

@@ -3,7 +3,6 @@ uuid = "b4b868b0-69a7-11e9-2db0-173b4e8e576c"
 version = "0.2.0"
 
 [deps]
-AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
 BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
 Dates = "ade2ca70-3891-5945-98fb-dc099432e06a"
 DocStringExtensions = "ffbed154-4ef7-542d-bbb7-c09d3a79fcae"
@@ -15,6 +14,5 @@ Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [compat]
-AbstractAlgebra = ">= 0.5.4"
 Nemo = ">= 0.14.3"
 julia = ">= 1.2.0"

docs/src/index.md

@@ -1426,6 +1426,9 @@ L013928
 V014107
 ```
 ```@docs
+L014606
+```
+```@docs
 L015128
 ```
 ```@docs
@@ -2194,6 +2197,9 @@ V257993
 L260884
 ```
 ```@docs
+V260884
+```
+```@docs
 L262071
 ```
 ```@docs
@@ -2281,15 +2287,27 @@ L291452
 L291973
 ```
 ```@docs
+V291973
+```
+```@docs
 L291974
 ```
 ```@docs
+V291974
+```
+```@docs
 L291975
 ```
 ```@docs
+V291975
+```
+```@docs
 L291976
 ```
 ```@docs
+V291976
+```
+```@docs
 L292222
 ```
 ```@docs

docs/src/modules.md

@@ -433,6 +433,18 @@ Return the numbers of partitions of an ``n``-set into ``m`` nonempty subsets.
 
 ## Set partitions of m-type
 
+For example consider the case n = 4. There are five integer partitions of 4:
+
+* P = [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]. The shapes are m times the parts of the integer partitions: S(m) = [[4m], [3m, m], [2m, 2m], [2m, m, m], [m, m, m, m]].
+
+* In the case m = 1 we look at set partitions of {1, 2, 3, 4} with sizes in  [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]] which gives rise to [1, 4, 3, 6, 1] with sum 15.
+
+* In the case m = 2 we look at set partitions of {1, 2, .., 8} with sizes in [[8], [6, 2], [4, 4], [4, 2, 2], [2, 2, 2, 2]] which gives rise to [1, 28, 35, 210, 105] with sum 379.
+
+* In the case m = 0 we look at set partitions of {} with sizes in [[0], [0, 0], [0, 0], [0, 0, 0], [0, 0, 0, 0]] which gives rise to [1, 1, 1, 1, 1] with sum 5 (because the only partition of the empty set is the set that contains the empty set, thus from the definition T(0,4) = Sum_{S(0)} card({0}) = A000041(4) = 5).
+
+* If n runs through 0, 1, 2,... then the result is an irregular triangle in which the n-th row lists multinomials for partitions of [m*n] which have only parts which are multiples of m. These are the triangles A080575 (m = 1), A257490 (m = 2), A327003 (m = 3), A327004 (m = 4). In the case m = 0 the triangle is A000012 subdivided into rows of length A000041. See the  references below how this case integrates into the full picture.
+
 | type  | m = 0 | m = 1 | m = 2 | m = 3 | m = 4 |
 |-------|-------|-------|-------|-------|-------|
 | by shape | [A000012](https://oeis.org/A000012) | [A036040](https://oeis.org/A036040) | [A257490](https://oeis.org/A257490) | [A327003](https://oeis.org/A327003) | [A327004](https://oeis.org/A327004) |

src/IntegerSequences.jl

@@ -2,8 +2,8 @@
 # Copyright Peter Luschny. License is MIT.
 # This file includes parts from Combinatorics.jl in modified form.
 
-# Version of: UTC 2019-10-23 14:30:02
-# d62119a0-f590-11e9-3b8f-97c1dfed0f02
+# Version of: UTC 2019-10-24 09:31:01
+# 3af04a00-f630-11e9-1ccb-3f8b137e1871
 
 # Do not edit this file, it is generated from the modules and will be overwritten!
 # Edit the modules in the src directory and build this file with BuildSequences.jl!
@@ -448,6 +448,7 @@ T011117,
 V011371,
 L013928,
 V014107,
+L014606,
 L015128,
 L023003,
 L023004,
@@ -660,7 +661,7 @@ V250283,
 V251568,
 V254749,
 V257993,
-L260884,
+L260884,V260884,
 L262071,
 T264428,
 T265606,
@@ -682,10 +683,10 @@ V281588,
 L290351,V290351,
 L291451,
 L291452,
-L291973,
-L291974,
-L291975,
-L291976,
+L291973,V291973,
+L291974,V291974,
+L291975,V291975,
+L291976,V291976,
 L292222,
 F293654,I293654,L293654,
 V295513,
@@ -7895,6 +7896,18 @@ SetNumber(n::Int) = Int(Nemo.bell(n))
 | central | [A053529](https://oeis.org/A053529) | [A210029](https://oeis.org/A210029) | [A281478](https://oeis.org/A281478) | [A281479](https://oeis.org/A281479) | [A281480](https://oeis.org/A281480) |
 
 
+For example consider the case n = 4. There are five integer partitions of 4:
+
+* P = [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]. The shapes are m times the parts of the integer partitions: S(m) = [[4m], [3m, m], [2m, 2m], [2m, m, m], [m, m, m, m]].
+
+* In the case m = 1 we look at set partitions of {1, 2, 3, 4} with sizes in  [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]] which gives rise to [1, 4, 3, 6, 1] with sum 15.
+
+* In the case m = 2 we look at set partitions of {1, 2, .., 8} with sizes in [[8], [6, 2], [4, 4], [4, 2, 2], [2, 2, 2, 2]] which gives rise to [1, 28, 35, 210, 105] with sum 379.
+
+* In the case m = 0 we look at set partitions of {} with sizes in [[0], [0, 0], [0, 0], [0, 0, 0], [0, 0, 0, 0]] which gives rise to [1, 1, 1, 1, 1] with sum 5 (because the only partition of the empty set is the set that contains the empty set, thus from the definition T(0,4) = Sum_{S(0)} card({0}) = A000041(4) = 5).
+
+* If n runs through 0, 1, 2,... then the result is an irregular triangle in which the n-th row lists multinomials for partitions of [m*n] which have only parts which are multiples of m. These are the triangles A080575 (m = 1), A257490 (m = 2), A327003 (m = 3), A327004 (m = 4). In the case m = 0 the triangle is A000012 subdivided into rows of length A000041. See the  references below how this case integrates into the full picture.
+
 | type  | m = 0 | m = 1 | m = 2 | m = 3 | m = 4 |
 |-------|-------|-------|-------|-------|-------|
 | by shape | [A000012](https://oeis.org/A000012) | [A036040](https://oeis.org/A036040) | [A257490](https://oeis.org/A257490) | [A327003](https://oeis.org/A327003) | [A327004](https://oeis.org/A327004) |
@@ -8288,79 +8301,135 @@ L005046(len) = [V005046(n) for n in 0:len-1]
 
 Return the number of set partitions of a 2n-set into even blocks which have even length minus the number of partitions into even blocks which have odd length.
 ```julia-repl
-julia> V260884(10)
-10-element Array{Nemo.fmpz,1}:
-??
+julia> V260884(19)
+5097857816569586800024019
 ```
 $(SIGNATURES)
 """
 V260884(n) = sum(AltEgfCoeffs(OrderedSetPolynomials(2, n)))
 """
 
----
+Return a list of the first ``len`` terms V260884(n).
+```julia-repl
+julia> L260884(10)
+10-element Array{Nemo.fmpz,1}:
+[1, -1, 2, -1, -43, 254, 4157, -70981, -1310398, 40933619]
+```
 $(SIGNATURES)
 """
 L260884(len) = [V260884(n) for n in 0:len-1]
 """
 
 Return the number of set partitions of type 3.
+```julia-repl
+julia> L291451(10)
+7-element Array{Nemo.fmpz,1}:
+[0, 1, 43690, 7128576, 99379280, 285885600, 190590400]
+```
 $(SIGNATURES)
 """
 L291451(n) = EgfCoeffs(OrderedSetPolynomials(3, n))
 """
 
-(3*n)! * [z^(3*n)] exp(exp(z)/3 + 2*exp(-z/2)*cos(z*sqrt(3)/2)/3 - 1).
+(3n)! [z^(3n)] exp(exp(z)/3 + 2exp(-z/2)cos(z sqrt(3)/2)/3 - 1).
+```julia-repl
+julia> V291973(9)
+31728742163212641
+```
 $(SIGNATURES)
 """
 V291973(n) = sum(EgfCoeffs(OrderedSetPolynomials(3, n)))
 """
 
----
+```julia-repl
+julia> L291973(6)
+6-element Array{Nemo.fmpz,1}:
+[1, 1, 11, 365, 25323, 3068521]
+```
 $(SIGNATURES)
 """
 L291973(len) = [V291973(n) for n in 0:len-1]
 """
 
-(3*n)! * [z^(3*n)] exp(-(exp(z)/3 + 2*exp(-z/2)*cos(z*sqrt(3)/2)/3 - 1)).
+(3n)! [z^(3n)] exp(-(exp(z)/3 + 2exp(-z/2) cos(z sqrt(3)/2)/3 - 1)).
+```julia-repl
+julia> V291974(9)
+-3166484321001
+```
 $(SIGNATURES)
 """
 V291974(n) = sum(AltEgfCoeffs(OrderedSetPolynomials(3, n)))
 """
 
----
+```julia-repl
+julia> L291974(6)
+6-element Array{Nemo.fmpz,1}:
+[1, -1, 9, -197, 6841, -254801]
+```
 $(SIGNATURES)
 """
 L291974(len) = [V291974(n) for n in 0:len-1]
 """
 
-exp(x*(cos(z) + cosh(z) - 2)/2)
+exp(x (cos(z) + cosh(z) - 2)/2)
+```julia-repl
+L291452(6)
+7-element Array{Nemo.fmpz,1}:
+[0, 1, 2098175, 2941884000, 181262956875, 1932541986375, 4509264634875]
+```
 $(SIGNATURES)
 """
 L291452(n) = EgfCoeffs(OrderedSetPolynomials(4, n))
 """
 
 Return ordered the number of set partitions of type 3.
+```julia-repl
+julia> V291975(9)
+926848347928901638652131
+```
 $(SIGNATURES)
 """
 V291975(n) = sum(EgfCoeffs(OrderedSetPolynomials(4, n)))
 """
 
----
+```julia-repl
+L291975(5)
+5-element Array{Nemo.fmpz,1}:
+[1, 1, 36, 6271, 3086331]
+```
 $(SIGNATURES)
 """
 L291975(len) = [V291975(n) for n in 0:len-1]
 """
 
 Return ordered the number of set partitions of type 4.
+```julia-repl
+julia> V291976(7)
+-6440372006517541
+```
 $(SIGNATURES)
 """
 V291976(n) = sum(AltEgfCoeffs(OrderedSetPolynomials(4, n)))
 """
 
----
+```julia-repl
+L291976(7)
+7-element Array{Nemo.fmpz,1}:
+[ 1, -1, 34, -5281, 2185429, -1854147586, 2755045819549]
+```
 $(SIGNATURES)
 """
 L291976(len) = [V291976(n) for n in 0:len-1]
+"""
+
+```julia-repl
+L014606(7)
+7-element Array{Nemo.fmpz,1}:
+[1, 1, 20, 1680, 369600, 168168000, 137225088000]
+```
+$(SIGNATURES)
+"""
+L014606(len) = Diagonal(n -> OrderedSetPolynomials(3, n), len)
 # *** StirlingLahNumbers.jl ****************
 """