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User:Peter Luschny/PermutationTrees
Contents
 1 Permutation Trees
 2 Counting permutation trees
 2.1 A123125 Permutation trees of power n and width k (Eulerian numbers).
 2.2 A179454 Permutation trees of power n and height k.
 2.3 A179455 Permutation trees of power n and height does not exceed k.
 2.4 A179456 Permutation trees of power n and height at most n − k.
 2.5 A179457 Permutation trees of power n and width does not exceed k.
 3 Three statistics on permutations
 4 Flattening the tree: the caterpillar notation
 5 The correspondence: Permutation <> Tree
 6 Summary: Three classifications of permutation trees.
 7 Number of maps p:[n]→[n] with p(x) ≤ x and p^[k](x) = p^[k1](x).
 8 The permutation trees with power 5
Permutation Trees
KEYWORDS: Permutation tree, rooted tree, classifications of permutation trees, Eulerian numbers, permutation, caterpillar notation.
Concerned with sequences: A008275, A008292, A179454, A179455, A179456, A179457.
Definitions
A permutation tree is a rooted tree that has vertex set {0,1,2,..,n} and root 0, and in which each child is larger than its parent and the children are in strict order from the left to the right. The power of a permutation tree is the number of descendants of the root. The height of a permutation tree is the number of descendants of the root on the longest chain starting at the root of the tree and ending at a leaf. The width of a permutation tree is the number of leafs.
The correspondence with the permutation is given by traversing the periphery of the tree starting at the right hand side of the root and recording a node whenever the node's right edge is passed.
For example below tree A has height 4 and width 1, the trees B, C, D all have height 3 and width 2, ..., and tree I has height 1 and width 4.
Example classifications
Permutations classified by height and by width of the associated rooted tree.
Trees are coded as parental lists.
Permutations and trees, n = 3.  


Permutations and trees, n = 4.  


Permutation trees with power 4
Counting permutation trees
A123125 Permutation trees of power n and width k (Eulerian numbers).
n\k  0  1  2  3  4  5  6  7  8  9 

0  1  
1  0  1  
2  0  1  1  
3  0  1  4  1  
4  0  1  11  11  1  
5  0  1  26  66  26  1  
6  0  1  57  302  302  57  1  
7  0  1  120  1191  2416  1191  120  1  
8  0  1  247  4293  15619  15619  4293  247  1  
9  0  1  502  14608  88234  156190  88234  14608  502  1 
A special case: A008292(n,2) = A000295(n) for n > 1.
A179454 Permutation trees of power n and height k.
n\k  0  1  2  3  4  5  6  7  8  9 

0  1  
1  0  1  
2  0  1  1  
3  0  1  4  1  
4  0  1  14  8  1  
5  0  1  51  54  13  1  
6  0  1  202  365  132  19  1  
7  0  1  876  2582  1289  265  26  1  
8  0  1  4139  19404  12859  3409  473  34  1  
9  0  1  21146  155703  134001  43540  7666  779  43  1 
Special cases: A179454(n,2) =
BellNumber(n)  1 = A058692(n) for n > 1;
A179454(n,n1) = A034856(n) for n > 1.
A179455 Permutation trees of power n and height does not exceed k.
Partial row sums of A179454.
n\k  0  1  2  3  4  5  6  7  8  9 

0  1  
1  0  1  
2  0  1  2  
3  0  1  5  6  
4  0  1  15  23  24  
5  0  1  52  106  119  120  
6  0  1  203  568  700  719  720  
7  0  1  877  3459  4748  5013  5039  5040  
8  0  1  4140  23544  36403  39812  40285  40319  40320  
9  0  1  21147  176850  310851  354391  362057  362836  362879  362880 
Special cases: A179455(n, 2) =
BellNumber(n) = A000110(n) for n > 1;
A179455(n, n1) = A033312(n) for n > 1.
A179456 Permutation trees of power n and height at most n − k.
Partial row sums of A179454 starting from the diagonal.
n\k  0  1  2  3  4  5  6  7  8  9 

0  1  
1  0  1  
2  0  2  1  
3  0  6  5  1  
4  0  24  23  9  1  
5  0  120  119  68  14  1  
6  0  720  719  517  152  20  1  
7  0  5040  5039  4163  1581  292  27  1  
8  0  40320  40319  36180  16776  3917  508  35  1  
9  0  362880  362879  341733  186030  52029  8489  823  44  1 
A special case: A179456(n, n1) = A000096(n).
A179457 Permutation trees of power n and width does not exceed k.
Partial row sums of A008292.
n\k  0  1  2  3  4  5  6  7  8  9 

0  1  
1  0  1  
2  0  1  2  
3  0  1  5  6  
4  0  1  12  23  24  
5  0  1  27  93  119  120  
6  0  1  58  360  662  719  720  
7  0  1  121  1312  3728  4919  5039  5040  
8  0  1  248  4541  20160  35779  40072  40319  40320  
9  0  1  503  15111  103345  259535  347769  362377  362879  362880 
A special case: A179457(n, 2) = A000325(n) for n > 1 (Grassmannian permutations).
Three statistics on permutations
The Combinatorial Statistic Finder
gives the definitions:
A combinatorial collection is a set with interesting combinatorial properties.
A combinatorial map is a combinatorially interesting map between combinatorial collections.
A combinatorial statistic is a combinatorially interesting map .
The combinatorial collection we want to consider here are permutations and we want to find a combinatorial statistic assigning a characteristic integer to every permutation. We assume that we already have a function which assignes to a rooted tree its height and a function which assignes to a rooted tree its width. Now we want to lift these characteristics to permutations.
The heightstatistic of permutations
In the first case we want to find a statistic and a map such that for all permutations p. This statistic and the map can be implemented in Sage/Python as:
def statistic(pi): if pi == []: return 0 h, i, branch, next = 0, len(pi), [0], pi[0] while true: while next < branch[1]: branch.pop() current = 0 while next > current: i = 1 h = max(h, len(branch)) if i == 0: return h branch.append(next) current, next = next, pi[i]
The distribution of the heights over the permutations now amounts to count the number of permutations with the same height.
def A179454_distribution(dim): for n in [0..dim]: L = [0]*(n+1) for p in Permutations(n): L[statistic(p)] += 1 print L
A call A179454_distribution(9) generates the above table A179454. The statistic can be found on FindStat as the statistic St000308.
The widthstatistic of permutations
The second case is the Eulerian case. Here we want to find a statistic and a map such that for all permutations p. This statistic and the map can be implemented in Sage/Python as:
def statistic_eulerian(pi): if pi == []: return 0 w, i, branch, next = 0, len(pi), [0], pi[0] while true: while next < branch[1]: branch.pop() current = 0 w += 1 while next > current: i = 1 if i == 0: return w branch.append(next) current, next = next, pi[i]
The distribution of widths over the permutations are counted by the Eulerian numbers. The function below computes a row in the Euler triangle.
def A123125_row(n): L = [0]*(n+1) for p in Permutations(n): L[statistic_eulerian(p)] += 1 return L [A123125_row(n) for n in range(7)]
The shapestatistic of permutations
The widthstatistic and the heightstatistic of permutations can be combined in a single statistic, the shapestatistic of permutations. It describes the shape of a permutation as the pair [width, height] of the associated permutation tree.
The small formal imprecision that a statistic is required to have the form can be easily overcome: We define st_shape(p) = 2^width(p)*3^height(p). There is no ambiguity in this encoding by the uniqueness of the prime factorization. Granted, there is an arbitrariness in the choice of 2 and 3 (any other two different prime numbers would do equally well) and in the order of width and height, but for the purpose of our exposition this encoding is a convenient way to represent and count the various shapes of permutation trees.
For example in the case n = 3 we have the statistic
[1, 2, 3] => (2, 2) => 36 [1, 3, 2] => (1, 3) => 54 [2, 1, 3] => (2, 2) => 36 [2, 3, 1] => (2, 2) => 36 [3, 1, 2] => (3, 1) => 24 [3, 2, 1] => (2, 2) => 36
Thus the 6 permutations can have three shapes [24, 36, 54] which have shape counts [1, 4, 1] respectively. Thus the permutations of {1, 2, 3} distribute in shape as [(24, 1), (36, 4), (54, 1)].
The number of different shapes for n ≥ 0 are
A265602 = 1, 1, 2, 3, 5, 7, 11, 14, 20, 25, 32, ...
We consider the fact that this sequence was not yet in the OEIS as a strong indication that this classification of permutations might be new.
The corresponding different shapes are
[(0,0)], [(1,1)], [(2,1), (1,2)], [(3,1), (2,2), (1,3)], [(2,2), (4,1), (3,2), (2,3), (1,4)], ...
or in encoded form
A265603 = [1], [6], [12, 18], [24, 36, 54], [36, 48, 72, 108, 162], [72, 96, 108, 144, 216, 324, 486], ...
The corresponding number of permutations with these shapes are
A265604 = [1], [1], [1, 1], [1, 4, 1], [3, 1, 11, 8, 1], [25, 1, 13, 26, 41, 13, 1], ...
A more formal description of the shapestatistic of permutations are bivariate polynomials defined as the sum of the monomials x^w*y^h where (w, h) is the shape of p, summed over all npermutations p.
For example in the case n = 4 we find
p(x,y) = x^4*y + 11*x^3*y^2 + 8*x^2*y^3 + x*y^4 + 3*x^2*y^2.
Looking at the coefficients of this polynomial we get the two lists
[0, y^4, 8*y^3 + 3*y^2, 11*y^2, y] [0, x^4, 11*x^3 + 3*x^2, 8*x^2, x]
Setting in turn y = 1 and x = 1 we arrive at the lists
[0, 1, 11, 11, 1] [0, 1, 14, 8, 1]
which are row 4 in triangle A123125 and triangle A179454, respectively.
In case n = 5 the reader might verify the polynomial
p(x,y) = x^5*y + 26*x^4*y^2 + 41*x^3*y^3 + 13*x^2*y^4 + x*y^5 + 25*x^3*y^2 + 13*x^2*y^3
This polynomial describes much the structure seen in the poster displayed at the bottom of this page. Looking at the coefficients we get:
[0, y^5, 13*y^4 + 13*y^3, 41*y^3 + 25*y^2, 26*y^2, y] [0, x^5, 26*x^4 + 25*x^3, 41*x^3 + 13*x^2, 13*x^2, x]
The Sage code for computing the shapestatistic is unsurprisingly almost identical to the scripts given above:
def statistic_shape(pi): if pi == []: return (0, 0) h, w, i, branch, next = 0, 0, len(pi), [0], pi[0] while true: while next < branch[1]: branch.pop() current = 0 w += 1 while next > current: i = 1 h = max(h, len(branch)) if i == 0: return (w, h) branch.append(next) current, next = next, pi[i]
This statistic can now be evaluated with regard to the different variables of interest for example with the function:
from sage.combinat.subset import list_to_dict def shape_row(n): y = var('y') S, L = 0, [] for p in Permutations(n): w, h = statistic_shape(p) S += x^w*y^h L.append(2^w*3^h) f = sorted(list_to_dict(L).items()) print " " print "*** n =", n, "***" print "dist of shapes: ", f print "shapes: ", [t[0] for t in f] print "shape counts: ", [t[1] for t in f] print "number of shapes: ", len(f) print "number of permutations: ", sum([t[1] for t in f]) print S print S.list(x) print S.list(y) for n in range(6): shape_row(n)
Flattening the tree: the caterpillar notation
The caterpillar traverses the periphery of the tree starting at the right hand side of the root and records a node whenever he passes the node's right edge.
The caterpillar notation is the usual one line notation of a permutation augmented by the symbol '^', inserted every time when the caterpillar is forced to climb up a level in the tree. Thus it is an one dimensional description of a permutation tree. The length of the caterpillar is twice the power of the tree.


Segments of the caterpillar which consist entirely of numbers are called down runs and segments consisting entirely of '^'s are called up runs of the caterpillar. A run is either a up run or a down run. The number of runs and their lengths are characteristics of a permutation.
The correspondence: Permutation <> Tree
Given a permutation, how to construct the associated rooted tree?
Maple code
perm2tree := proc(perm) local tree, i, next, root, branch, current; tree := []; root := 0; branch := [root]; i := 1; next := perm[i]; while true do while next < branch[nops(branch)] do branch := subsop(nops(branch) = NULL, branch); od; current := root; while next > current do branch := [op(branch),next]; i := i + 1; if i > nops(perm) then tree := [op(tree),branch]; RETURN(tree) fi; current := next; next := perm[i]; od; tree := [op(tree),branch]; od end:
An example use is:
ListPermTrees := proc(n) local p, P; P := combinat[permute](n); for p in P do lprint(p, "=>", perm2tree(p)); od end: ListPermTrees(4);
SageMath code
def plot_tree(E): G = Graph() G.add_edges(E) G.show(layout='tree', tree_root=0, tree_orientation='down') def permutation_to_tree(pi): if pi == []: return [[0]] root, i, next = 0, 0, pi[0] branch, edges = [root], [] caterpillar = "" while true: while next < branch[1]: branch.pop() caterpillar += "^" current = root while next > current: edges.append([branch[1], next]) caterpillar += str(next) branch.append(next) i += 1 if i == len(pi): l = 2*len(pi)len(caterpillar) caterpillar += "^" * l return caterpillar #plot_tree(edges) #return edges current, next = next, pi[i] for n in (1..4): for p in Permutations(n): print p, "=>", permutation_to_tree(p)
[1, 2, 3, 4] => 1234^^^^ [1, 2, 4, 3] => 124^3^^^ [1, 3, 2, 4] => 13^24^^^ [1, 3, 4, 2] => 134^^2^^ [1, 4, 2, 3] => 14^23^^^ [1, 4, 3, 2] => 14^3^2^^ [2, 1, 3, 4] => 2^134^^^ [2, 1, 4, 3] => 2^14^3^^ [2, 3, 1, 4] => 23^^14^^ [2, 3, 4, 1] => 234^^^1^ [2, 4, 1, 3] => 24^^13^^ [2, 4, 3, 1] => 24^3^^1^ [3, 1, 2, 4] => 3^124^^^ [3, 1, 4, 2] => 3^14^2^^ [3, 2, 1, 4] => 3^2^14^^ [3, 2, 4, 1] => 3^24^^1^ [3, 4, 1, 2] => 34^^12^^ [3, 4, 2, 1] => 34^^2^1^ [4, 1, 2, 3] => 4^123^^^ [4, 1, 3, 2] => 4^13^2^^ [4, 2, 1, 3] => 4^2^13^^ [4, 2, 3, 1] => 4^23^^1^ [4, 3, 1, 2] => 4^3^12^^ [4, 3, 2, 1] => 4^3^2^1^
Given a rooted tree, how to construct the associated permutation?
Maple code
We assume the tree given by the list of its branches, this means as the list of all the chains starting at the root of the tree and ending at a leaf.
tree2perm := proc(tree) local j, k, perm, branch, leaf; perm := []; for j from 1 to nops(tree) do branch := tree[j]; for k from 2 to nops(branch) do leaf := branch[k]; if not member(leaf, perm) then perm := [op(perm), leaf] fi; od; od: perm end:
An example use is:
ListPerms := proc(n) local p,P,pi,t; P := combinat[permute](n); for p in P do t := perm2tree(p): pi := tree2perm(t); lprint(t, "=>", pi); od end: ListPerms(4);
SageMath code
def tree_to_permutation(tree): perm = tree.translate(None, '^') return [int(p) for p in perm] for n in (1..4): for p in Permutations(n): t = permutation_to_tree(p) print t, "=>", tree_to_permutation(t)
1234^^^^ => [1, 2, 3, 4] 124^3^^^ => [1, 2, 4, 3] 13^24^^^ => [1, 3, 2, 4] 134^^2^^ => [1, 3, 4, 2] 14^23^^^ => [1, 4, 2, 3] 14^3^2^^ => [1, 4, 3, 2] 2^134^^^ => [2, 1, 3, 4] 2^14^3^^ => [2, 1, 4, 3] 23^^14^^ => [2, 3, 1, 4] 234^^^1^ => [2, 3, 4, 1] 24^^13^^ => [2, 4, 1, 3] 24^3^^1^ => [2, 4, 3, 1] 3^124^^^ => [3, 1, 2, 4] 3^14^2^^ => [3, 1, 4, 2] 3^2^14^^ => [3, 2, 1, 4] 3^24^^1^ => [3, 2, 4, 1] 34^^12^^ => [3, 4, 1, 2] 34^^2^1^ => [3, 4, 2, 1] 4^123^^^ => [4, 1, 2, 3] 4^13^2^^ => [4, 1, 3, 2] 4^2^13^^ => [4, 2, 1, 3] 4^23^^1^ => [4, 2, 3, 1] 4^3^12^^ => [4, 3, 1, 2] 4^3^2^1^ => [4, 3, 2, 1]
Summary: Three classifications of permutation trees.
Proposition 1. A permutation p has k down runs if and only if the permutation tree T(p) has width k.
Proposition 2. The length of the longest run of a permutation p is k if and only if the permutation tree T(p) has height k.
Proposition 1 is covered by the enumeration of permutations by the Eulerian numbers A008292 and proposition 2 by the enumeration of permutations by A179454.
The power of the caterpillar notation becomes even more obvious when we note that it immediately leads to a third classification. Just look at the tail of the caterpillar. Classifying the caterpillar by the length of the last up run gives the Stirling cycle numbers! (Definition as in CM, table 245, or DLMF, 26.13.3., ABS(A008275).) This third classification is equivalent to:
Proposition 3. Stirling cycle numbers classify permutation trees according to the length of the first (leftmost) branch.
This description is somewhat simpler then the standard definition using cycle arrangements.
Number of maps p:[n]→[n] with p(x) ≤ x and p^[k](x) = p^[k1](x).
Joerg Arndt gave this interpretation in A187761 and indicated an algorithm for the generation with restricted growth strings (RGS) for maps. A C# implementation of this algorithm below:
// Generating algorithm due to Joerg Arndt. // Restricted growth strings (RGS) for maps. // A000110: f: [n] > [n] with f(x) <= x and f(x) = f(f(x)). // A187761: f: [n] > [n] with f(x) <= x and f(f(x)) = f(f(f(x))) // general: kth column of A179455 // f: [n] > [n] with f(x) <= x and f^[k1](x) = f^[k](x) namespace OEIS { class A179455 { static int[] a; static void Main() { for (int n = 1; n < 11; n++) { var count = row(n); Console.Write("[n = " + n + "] "); for (int i = 0; i < n; i++) Console.Write(count[i] + ","); Console.WriteLine(); } Console.ReadLine(); } static int[] row(int n) { int[] count = new int[n]; for (int k = 0; k < n; k++) { a = new int[n]; do { count[k]++; } while (generate(n, k) != 0); } return count; } static int generate(int n, int k) { if (n == 0  k == 0) return 0; int j = n, f, f1, fl; while (j != 0) { f = a[j]; while (++f <= j) { a[j] = f; f1 = f; fl = f1; for (int i = 0; i < k; i++) { fl = f1; f1 = a[fl]; } if (f1 == fl) return j; } a[j] = 0; } return 0; }}}

The image of the blackred caterpillar was made
by Jill Chen from Taipei, Taiwan. It was down loaded from commons.wikimedia.org and is
licensed under the Creative Commons Attribution 2.0 Generic license.
The permutation trees with power 5
You can download this poster of permutation trees as a pdffile. And this article as a pdffile.