tree

Author: PeterLuschny

Project.toml

@@ -3,7 +3,6 @@ uuid = "b4b868b0-69a7-11e9-2db0-173b4e8e576c"
 version = "0.2.0"
 
 [deps]
-Atom = "c52e3926-4ff0-5f6e-af25-54175e0327b1"
 BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
 DataStructures = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8"
 Dates = "ade2ca70-3891-5945-98fb-dc099432e06a"
@@ -12,7 +11,6 @@ Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
 HTTP = "cd3eb016-35fb-5094-929b-558a96fad6f3"
 InteractiveUtils = "b77e0a4c-d291-57a0-90e8-8db25a27a240"
 IterTools = "c8e1da08-722c-5040-9ed9-7db0dc04731e"
-Juno = "e5e0dc1b-0480-54bc-9374-aad01c23163d"
 Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 

docs/src/index.md

@@ -772,15 +772,27 @@ V000045
 is000045
 ```
 ```@docs
+V000055
+```
+```@docs
 G000073
 ```
 ```@docs
 L000073
 ```
 ```@docs
+V000081
+```
+```@docs
 L000085
 ```
 ```@docs
+V000088
+```
+```@docs
+V000106
+```
+```@docs
 G000108
 ```
 ```@docs
@@ -1057,6 +1069,9 @@ L002448
 L002476
 ```
 ```@docs
+V002494
+```
+```@docs
 L002808
 ```
 ```@docs
@@ -1549,6 +1564,9 @@ V035327
 L036038
 ```
 ```@docs
+L036039
+```
+```@docs
 L036040
 ```
 ```@docs
@@ -1645,6 +1663,15 @@ T053121
 V054248
 ```
 ```@docs
+V055542
+```
+```@docs
+V055543
+```
+```@docs
+V055544
+```
+```@docs
 V055634
 ```
 ```@docs
@@ -1801,9 +1828,15 @@ T084938
 V086799
 ```
 ```@docs
+L087803
+```
+```@docs
 L088218
 ```
 ```@docs
+V088887
+```
+```@docs
 T088969
 ```
 ```@docs
@@ -1837,6 +1870,9 @@ V094587
 T094665
 ```
 ```@docs
+V095350
+```
+```@docs
 V095794
 ```
 ```@docs
@@ -1939,6 +1975,9 @@ L111533
 T111593
 ```
 ```@docs
+L115621
+```
+```@docs
 T116392
 ```
 ```@docs
@@ -2110,6 +2149,9 @@ I206942
 L206942
 ```
 ```@docs
+V209397
+```
+```@docs
 L210029
 ```
 ```@docs
@@ -2131,6 +2173,9 @@ T216916
 V216919
 ```
 ```@docs
+V217420
+```
+```@docs
 T217537
 ```
 ```@docs
@@ -2469,3 +2514,6 @@ V327987
 ```@docs
 L327988
 ```
+```@docs
+L328917
+```

docs/src/modules.md

@@ -248,6 +248,8 @@ The partition numbers and the number of partitions of n into k parts are given a
 
 The sum of all partition coefficients of n is efficiently computed with L005651.
 
+* V000041, V088887, I072233, L072233, L036038, L078760, L005651, L262071, L292222, L115621, L328917
+
    🔶  [JacobiTheta](https://github.com/OpenLibMathSeq/IntegerSequences.jl/blob/master/src/JacobiTheta.jl)
 
 
@@ -288,6 +290,12 @@ For background information see
 * J.-P. Allouche, T. Johnson, [Narayana's Cows and Delayed Morphisms](http://recherche.ircam.fr/equipes/repmus/jim96/actes/Allouche.ps).
 * C.M. Wilmott, [From Fibonacci to the mathematics of cows and quantum circuitry](https://iopscience.iop.org/article/10.1088/1742-6596/574/1/012097/pdf).
 
+   🔶  [NodesAndEdges](https://github.com/OpenLibMathSeq/IntegerSequences.jl/blob/master/src/NodesAndEdges.jl)
+
+Rooted trees and similar topics.
+
+* V000055, V000081, V000106, V209397, V217420, V095350, V002494, V055542, V055543, V055544, V000088, L087803, L036039
+
    🔶  [NumberTheory](https://github.com/OpenLibMathSeq/IntegerSequences.jl/blob/master/src/NumberTheory.jl)
 
 
@@ -424,18 +432,6 @@ Return the numbers of partitions of an ``n``-set into ``m`` nonempty subsets.
    🔶  [SetPartitionsMType](https://github.com/OpenLibMathSeq/IntegerSequences.jl/blob/master/src/SetPartitionsMType.jl)
 
 
-
-## Ordered set partitions of m-type
-
-| type | m = 0 | m = 1 | m = 2 | m = 3 | m = 4 |
-|----|-------|-------|-------|-------|-------|
-| by shape | [A178803](https://oeis.org/A178803) | [A133314](https://oeis.org/A133314) | [A327022](https://oeis.org/A327022) | [A327023](https://oeis.org/A327023) | [A327024](https://oeis.org/A327024) |
-| by length | [A318144](https://oeis.org/A318144) | [A131689](https://oeis.org/A131689) | [A241171](https://oeis.org/A241171) | [A278073](https://oeis.org/A278073) | [A278074](https://oeis.org/A278074) |
-| diagonal | [A000142](https://oeis.org/A000142) | [A000142](https://oeis.org/A000142) | [A000680](https://oeis.org/A000680) | [A014606](https://oeis.org/A014606) | [A014608](https://oeis.org/A014608) |
-| row sum | [A101880](https://oeis.org/A101880) | [A000670](https://oeis.org/A000670) | [A094088](https://oeis.org/A094088) | [A243664](https://oeis.org/A243664) | [A243665](https://oeis.org/A243665) |
-| alt row sum | [A260845](https://oeis.org/A260845) | [A033999](https://oeis.org/A033999) | [A028296](https://oeis.org/A028296) | [A002115](https://oeis.org/A002115) | [A211212](https://oeis.org/A211212) |
-| 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) |
-
 ## Set partitions of m-type
 
 For example consider the case n = 4. There are five integer partitions of 4:
@@ -461,6 +457,17 @@ For example consider the case n = 4. There are five integer partitions of 4:
 
 See also [A260876](https://oeis.org/A260876).
 
+## Ordered set partitions of m-type
+
+| type | m = 0 | m = 1 | m = 2 | m = 3 | m = 4 |
+|------|-------|-------|-------|-------|-------|
+| by shape | [A178803](https://oeis.org/A178803) | [A133314](https://oeis.org/A133314) | [A327022](https://oeis.org/A327022) | [A327023](https://oeis.org/A327023) | [A327024](https://oeis.org/A327024) |
+| by length | [A318144](https://oeis.org/A318144) | [A131689](https://oeis.org/A131689) | [A241171](https://oeis.org/A241171) | [A278073](https://oeis.org/A278073) | [A278074](https://oeis.org/A278074) |
+| diagonal | [A000142](https://oeis.org/A000142) | [A000142](https://oeis.org/A000142) | [A000680](https://oeis.org/A000680) | [A014606](https://oeis.org/A014606) | [A014608](https://oeis.org/A014608) |
+| row sum | [A101880](https://oeis.org/A101880) | [A000670](https://oeis.org/A000670) | [A094088](https://oeis.org/A094088) | [A243664](https://oeis.org/A243664) | [A243665](https://oeis.org/A243665) |
+| alt row sum | [A260845](https://oeis.org/A260845) | [A033999](https://oeis.org/A033999) | [A028296](https://oeis.org/A028296) | [A002115](https://oeis.org/A002115) | [A211212](https://oeis.org/A211212) |
+| 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) |
+
    🔶  [SpigotPi](https://github.com/OpenLibMathSeq/IntegerSequences.jl/blob/master/src/SpigotPi.jl)
 
 Computes the first n decimal digits of Pi, uses a variant of the spigot algorithm valid as long as the number of digits <= 54900. Based on ideas of A. Sale (1968). Algorithm due to D. Saada (1988) and S. Rabinowitz (1991). Proof due to [Rabinowitz and S. Wagon](https://www.maa.org/sites/default/files/pdf/pubs/amm_supplements/Monthly_Reference_12.pdf) (1995).

src/IntPartitions.jl

@@ -5,13 +5,18 @@
 
 module IntPartitions
 
-using GeneralizedBinomial
+using GeneralizedBinomial, DataStructures
 
 export ModuleIntPartitions
 export IntegerPartitions, PartitionNumber
 export PartOrder, byNumPart, byMaxPart
 export PartitionCoefficientsByLength, PartitionCoefficientsByBiggestPart
 export V000041, I072233, L072233, L036038, L078760, L005651, L262071, L292222
+export L115621, L328917, V088887
+
+#=
+a(n) is the sum of the parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(33) = 7 because the partition with Heinz number 33 = 3 * 11 is [2,5]. - Emeric Deutsch, May 19 2015
+=#
 
 """
 
@@ -27,6 +32,8 @@ The partition coefficients, which are the multinomial coefficients applied to pa
 The partition numbers and the number of partitions of n into k parts are given as PartitionNumber(n) and PartitionNumber(n, k), (V000041, L072233).
 
 The sum of all partition coefficients of n is efficiently computed with L005651.
+
+* V000041, V088887, I072233, L072233, L036038, L078760, L005651, L262071, L292222, L115621, L328917
 """
 const ModuleIntPartitions = ""
 
@@ -287,6 +294,32 @@ function L292222(n)
     [sum(Multinomial(p) for p ∈ IntegerPartitions(n, k)) for k ∈ 1:n]
 end
 
+"""
+Return the signature of partitions in Hindenburg order.
+"""
+function L115621(n)
+    h(p) = sort(collect(values(counter(p))))
+    [h(p) for p ∈ IntegerPartitions(n)]
+end
+
+"""
+Return the types of signatures of partitions of n, ordered firstly by decreasing greatest parts, then decreasing sum of parts, then by increasing number of parts.
+"""
+function L328917(n)
+    n == 0 && return [[0]]
+    h(p) = sort(collect(values(counter(p))), rev=true)
+    sort(unique([h(p) for p ∈ IntegerPartitions(n)]), rev=true)
+end
+
+"""
+Return the number of types of signatures of partitions of n.
+"""
+function V088887(n)
+    h(p) = sort(collect(values(counter(p))), rev=true)
+    length(unique([h(p) for p ∈ IntegerPartitions(n)]))
+end
+
+
 #START-TEST-########################################################
 
 using Test, SeqUtils
@@ -421,6 +454,17 @@ function demo()
     println("\n-- Sum of all partition coefficients of n")
     L005651(10) |> println
     println()
+
+    println("\n-- Signature of partitions in Hindenburg order.")
+    for n ∈ 1:6
+        L115621(n) |> Println
+    end
+
+    println("\n-- Types of partitions (counted by V088887).")
+    for n ∈ 0:6
+        print(n, " : "); Println(L328917(n))
+    end
+    println()
 end
 
 """