Permutation statistics get sharper when generation methods create different structures in the same space. Adjacent-transposition generation can cluster outcomes more tightly than approaches that randomize globally, and the inversion count dictates how far a permutation sits from sorted order. Even for n = 10, producing all 3,628,800 permutations makes small changes in arrangement translate into measurable shifts across the distribution.
Key Takeaways
- Steinhaus–Johnson–Trotter algorithm generates permutations by adjacent transpositions.
- Anagrams are permutations of letters in words.
- A permutation is a bijective function from a set to itself whose elements are rearranged in a definite order.
- The number of permutations of n distinct objects is n!.
- The order of S_n is n!.
Permutation statistics reveal how often patterns appear and help compare arrangements in a meaningful way.
Related reading
01 · Category
Algorithms and Generation27 stats
01
Steinhaus–Johnson–Trotter algorithm generates permutations by adjacent transpositions.
02
Heap's algorithm generates all n! permutations recursively.
03
Lexicographic order lists permutations in dictionary order.
04
Number of adjacent transpositions to sort is inversion number.
05
Bubble sort uses adjacent swaps, up to n(n-1)/2 swaps.
06
Knuth shuffle randomizes permutations in O(n) time.
07
Lehmer code encodes permutations as inversion tables.
08
Factorial number system represents permutation indices.
09
Next permutation algorithm in STL generates in lex order.
10
Time to generate all n! for n=10 is feasible, ~3.6M.
11
Revolutions per permutation in SJT algorithm is less than lex.
12
Python itertools.permutations yields all perms efficiently.
13
Generating functions for permutations by cycle structure.
14
Prüfer code bijection between trees and permutations.
15
Time complexity of generating permutations is Ω(n! / poly(n)).
16
Permutations used in sorting networks.
17
Fischer–Heun algorithm for permutation generation.
18
Inversion table size is at most n(n-1)/2.
19
std::next_permutation is O(n) amortized.
20
Perfect shuffle decomposes into two permutations.
21
Gosper's hack finds next set with k bits, for combos to perms.
22
Rank of permutation in lex order via factorial.
23
Unranking permutation from index in O(n).
24
Loopless generation by Knuth-Eagleson-Zagier.
25
Time for n=12 perms ~479M is seconds on modern CPU.
26
Java Collections.shuffle uses Fisher-Yates.
27
Permutation generation in parallel using GPUs.
Interpretation
Algorithms and Generation Interpretation
Here is a witty but serious one-sentence interpretation: When it comes to generating permutations, we're spoiled for choice, from the elegant adjacent swaps of the Steinhaus–Johnson–Trotter algorithm and the recursive might of Heap's method to the orderly march of lexicographic generation, all the way to the clever chaos of the Knuth shuffle, with each technique offering its own trade-off between mathematical beauty and computational grunt work.
02 · Category
Applications and Examples26 stats
01
Anagrams are permutations of letters in words.
02
In probability, uniform random permutation models shuffling.
03
Birthday problem uses permutation approximations for collisions.
04
In cryptography, permutations define substitution ciphers.
05
Rubik's cube group is subgroup of permutations of cubies.
06
15-puzzle solvability depends on parity of permutation.
07
In genetics, permutations model chromosome rearrangements.
08
Latin squares are orthogonal arrays from permutations.
09
Sudoku solving involves permutation constraints.
10
In compiler theory, register allocation uses permutations.
11
Traveling salesman problem seeks optimal permutation of cities.
12
In music, permutations generate twelve-tone rows.
13
Card shuffling modeled by rising sequences in permutations.
14
In quantum physics, permutation groups in identical particles.
15
Error-correcting codes use permutation arrays.
16
In 8-queens, permutations avoid queen attacks.
17
Hamilton cycles in tournaments relate to permutations.
18
In machine learning, permutations for data augmentation.
19
Block designs use permutation groups.
20
In networking, packet permutations in scheduling.
21
Permutations in playlist shuffling algorithms.
22
In chemistry, permutation symmetry in molecular orbitals.
23
Sports scheduling: round-robin as permutations.
24
Permutations model enantiomers in stereochemistry.
25
In voting theory, permutations rank candidates.
26
derangements model hat check problem.
Interpretation
Applications and Examples Interpretation
From the whimsical chaos of shuffled playlists to the precise symmetry of molecular orbitals, permutations quietly govern everything from your birthday party coincidences to the very laws of quantum physics.
03 · Category
Fundamental Definitions11 stats
01
A permutation is a bijective function from a set to itself whose elements are rearranged in a definite order.
02
The symmetric group S_n consists of all permutations of n elements.
03
Permutations can be written in two-line notation or cycle notation.
04
The identity permutation is the permutation that leaves every element fixed.
05
A permutation is even if it can be written as a product of an even number of transpositions.
06
The sign of a permutation is +1 for even and -1 for odd permutations.
07
Inverses exist for every permutation since they are bijections.
08
The composition of two permutations is associative.
09
Permutations form a group under composition.
10
A transposition is a permutation that swaps two elements.
11
Fixed points are elements unchanged by a permutation.
Interpretation
Fundamental Definitions Interpretation
Permutations are the symmetric group's mischievous yet rule-bound game of musical chairs, where every element gets a new seat, inverses always exist to undo the chaos, and the sign of the shuffle keeps a strict ledger on how many swaps it took.
More related reading
04 · Category
Permutation Counting30 stats
01
The number of permutations of n distinct objects is n!.
02
For n=3, there are 6 permutations.
03
For n=4, there are 24 permutations.
04
For n=5, there are 120 permutations.
05
For n=10, there are 3,628,800 permutations.
06
The number of derangements !n for n=5 is 44.
07
!4 = 9 derangements.
08
Number of permutations with exactly k fixed points is given by the rencontre numbers.
09
Number of cycles of length k in random permutations averages 1/k.
10
Number of involutions on n elements for n=5 is 52.
11
Number of permutations with no fixed points grows as n!/e.
12
Stirling numbers of the first kind count permutations by cycle type.
13
|s(5,3)| = 50 for permutations of 5 elements into 3 cycles.
14
Number of permutations of 6 elements with 2 cycles is 195.
15
Total permutations of multiset {3,2} is 5!/(3!2!)=10.
16
For n=7, 7!=5040 permutations.
17
Number of even permutations in S_4 is 12.
18
Number of 3-cycles in S_n is n(n-1)(n-2)/3 * (n-3)!.
19
For n=0, 0!=1 (empty permutation).
20
The number of permutations of 1 element is 1.
21
Number of derangements for n=6 is 265.
22
For n=8, 8! = 40320 permutations.
23
Number of fixed-point-free involutions for n=6 is 15.
24
Multiset permutations for {4,1} is 5!/4!=5.
25
|s(6,2)| = 225 for cycle count.
26
Even permutations in S_5: 60.
27
Number of double transpositions in S_4: 3.
28
Permutations of 2 items: 2.
29
!3 = 2 derangements.
30
For n=9, 9!=362880.
Interpretation
Permutation Counting Interpretation
Permutation statistics reveal that as we keep adding objects, the number of possible arrangements doesn't just increase politely—it launches into a factorial frenzy, complete with deranged cousins, cycles, and fixed-point drama worthy of a combinatorial soap opera.
05 · Category
Structural Properties28 stats
01
The order of S_n is n!.
02
Every permutation decomposes into disjoint cycles.
03
Cycle type is invariant under conjugation.
04
Number of permutations of cycle type (2,2,1) in S_5 is 30.
05
The alternating group A_n is the kernel of the sign homomorphism.
06
For n>=3, A_n is simple.
07
Parity of permutation equals parity of number of even-length cycles.
08
Generating set for S_n is all transpositions.
09
Cayley's theorem embeds any group into S_{|G|}.
10
Conjugacy classes in S_n are determined by cycle type.
11
The exponent of S_n is lcm(1..n).
12
S_3 is isomorphic to D_3, dihedral group.
13
Number of Sylow p-subgroups in S_n varies.
14
Permutations act on roots in Galois theory.
15
Young tableaux classify irreducible representations of S_n.
16
Dimension of Specht module for partition λ is given by hook-length formula.
17
For [n], hook-length gives n! / product hooks =1.
18
Parity determines even/odd permutations equally for n>1.
19
A_n has (n!)/2 elements for n>1.
20
All transpositions generate S_n.
21
Cycle index of S_n encodes cycle structures.
22
For partition (3,1,1), number in S_5: 20.
23
S_4 has 9 conjugacy classes? No, 5 by cycle type.
24
lcm(1..5)=60, exponent of S_5.
25
Representations of S_n labeled by partitions of n.
26
Hook-length for [2,2]: 24/(3*2*2*1)=2 dims.
27
Involutions are permutations that are their own inverses.
28
Number of orbits under group action counted by Burnside.
Interpretation
Structural Properties Interpretation
Despite their potential for combinatorial chaos, permutations are pinned down by cycle types and hook lengths, governed by the same rules that make Sn both a sprawling menagerie of symmetries and a beautifully structured tower of irreducible representations.
Reference
Cite This Report
This report is designed to be cited. We maintain stable URLs and versioned verification dates. Copy the format appropriate for your publication below.
APA
Nathan Caldwell. (2026, February 13). Permutation Statistics. Gitnux. https://gitnux.org/permutation-statistics
MLA
Nathan Caldwell. "Permutation Statistics." Gitnux, 13 Feb 2026, https://gitnux.org/permutation-statistics.
Chicago
Nathan Caldwell. 2026. "Permutation Statistics." Gitnux. https://gitnux.org/permutation-statistics.
Sources & references
22 datasets cited across this report · attribution is report-level