Did you know that arranging just twenty-five distinct objects yields more possible orders than there are stars in the observable universe?
Key Takeaways
- The number of permutations of 1 element is 1
- The number of permutations of 2 elements is 2
- The number of permutations of 3 elements is 6
- The symmetric group S_3 has 6 elements, all permutations of 3 objects
- The order of the symmetric group S_n is n! for any n
- S_4 has 24 elements, isomorphic to rotations of tetrahedron plus reflections
- Number of derangements !3 = 2 in S_3
- Number of derangements !4 = 9
- Number of derangements !5 = 44
- Number of cycles of length k in random permutation is Harmonic number approx 1/k
- Expected number of fixed points (1-cycles) in random S_n is 1
- Probability a random permutation is a single n-cycle is (n-1)! / n! = 1/n
- The number of permutations of n objects with exactly k fixed points is the rencontre number D(n,k)
- In probability, the birthday problem uses 1 - P(no collision) ≈ 1 - e^{-n^2/2m}, related to partial derange
- Sorting by reversals: minimal number for random perm is approx n ln n / ln 2 or something, but pancake sorting depth 15/8 n+
Permutation counts grow factorially and quickly become extremely large.
Applications
- The number of permutations of n objects with exactly k fixed points is the rencontre number D(n,k)
- In probability, the birthday problem uses 1 - P(no collision) ≈ 1 - e^{-n^2/2m}, related to partial derange
- Sorting by reversals: minimal number for random perm is approx n ln n / ln 2 or something, but pancake sorting depth 15/8 n+
- In cryptography, number of S-boxes in DES is 8, each 4x4 to 4 bit, nonlinear perms
- Steinhaus–Johnson–Trotter algorithm generates all n! perms by adjacent transpositions, n! -1 swaps
- Heap's algorithm generates perms recursively, efficient in practice
- In quantum computing, perm group used in some circuits, but S_n reps for error correction
- Number of ways to arrange n distinct books on shelf: n!
- In round-robin tournament, number of ways to schedule: (2n-1)!! * n! or related perms
- Lehmer code encodes perms as inversion tables 0 to n!-1
- In genome rearrangements, breakpoint graph uses cycles in perm graph
- Number of Latin squares of order n related to reduced perms, but #LS(n) ~ n! (n/e)^ {n^2} exp
- In design theory, mutually orthogonal Latin squares from finite fields, perms involved
- Fast Fourier transform over S_n uses Young tableaux, complexity O(n^2 log n) or better
- In machine learning, permutation invariance in equivariant nets, stats on equiv classes
- Traveling salesman problem: n! /2 tours for complete graph
- In compiler optimization, instruction scheduling as list scheduling perms
- Rubik's cube group is subgroup of wreath product, 43 quintillion positions ~ perms of pieces
- DNA microarray experiments use perm tests for significance
- In voting theory, number of rankings n!, Condorcet paradox prob 89% for n=3
- Network routing: OSPF uses shortest path, but perm of packets
- Time to generate all perms with std::next_permutation in C++ is O(n n!)
- In chemistry, stereoisomers from chiral centers: 2^k but perms for indistinct
Applications Interpretation
Permutations are the mathematical glue holding together everything from our chance of sharing a birthday to the mind-boggling number of ways to scramble a Rubik's Cube, proving that while order matters, the chaos of possibilities is what truly makes the world—and math—so fascinating.
Cycle Structures
- Number of cycles of length k in random permutation is Harmonic number approx 1/k
- Expected number of fixed points (1-cycles) in random S_n is 1
- Probability a random permutation is a single n-cycle is (n-1)! / n! = 1/n
- Number of permutations with cycle type partition λ of n is n! / (Π k^{m_k} m_k! )
- Number of 2-cycles (transpositions) in S_n is C(n,2)
- Average number of cycles in random permutation is H_n ≈ ln n + γ
- Probability of being even permutation is 1/2 for n>1
- Number of permutations with exactly k cycles is |s(n,k)| * k!, unsigned Stirling1
- For cycle type (n), number is (n-1)!
- Expected length of longest cycle is approx li(n) or ~ n / e^γ log log n wait, actually ~c n for some, but detailed: asymptotic ρ n where ρ root of eq
- Number of involutions (cycles ≤2) in S_n is sum over k C(n,2k) * (2k)! / (2^k k!) * 1/(n-2k)!
- Probability two random perms commute is sum 1/c(λ)^2 over partitions
- Cycle index of S_n is average of x_1^{c1}... over perms
- Number of 3-cycles: [n(n-1)(n-2)] / 3 *2? Wait n(n-1)(n-2)/3, no (n choose 3)*2
- In S_5, number of cycle type (3,2): 5! / (3*2 *1!1!) = 20
- Generating function for number of permutations by cycles is exp( sum x^k /k )
- Variance of number of k-cycles is 1/k approx
- Probability of cycle type with m_k cycles of len k is approx product (1/k)^{m_k} / m_k!
- Number of fixed-point-free involutions for even n: (n-1)!!
- In random perm, prob of having a k-cycle containing specific point is 1/k for k=1 to n
- Total number of cycles across all perms: sum |s(n,k)| k! * k = n! H_n
- For S_10, number of cycle type (4,3,2,1): 10! / (4*3*2*1 *1!1!1!1!) = 6300* something wait 10!/ (4*3*2*1) = 94500
- Ewens distribution models cycle structure with parameter θ
- Number of permutations that are products of two n-cycles: complicated, but stats exist
Cycle Structures Interpretation
When juggling permutations, expect chaos in a predictable order: while one fixed point is typical and single cycles occur about once every n tries, the harmony of cycles—governed by Stirling’s sneaky numbers and generating functions—ensures that on average we’ll find roughly the logarithm of our troubles plus Euler’s constant, reminding us that even within random disorder, combinatorial truth prevails.
Derangements
- Number of derangements !3 = 2 in S_3
- Number of derangements !4 = 9
- Number of derangements !5 = 44
- Number of derangements !6 = 265
- Number of derangements !7 = 1854
- Number of derangements !8 = 14833
- Number of derangements !9 = 133496
- Number of derangements !10 = 1334961
- The probability of a random permutation being a derangement approaches 1/e ≈ 0.367879 as n→∞
- Exact formula for !n = n! * Σ_{k=0}^n (-1)^k / k!
- Number of derangements with exactly one fixed point is C(n,1) * !(n-1)
- !0 = 1, !1 = 0
- Recurrence: !n = (n-1) [!(n-1) + !(n-2)]
- !11 = 14636212681
- Number of permutations with no fixed points in S_12 is 176214841
- Inclusion-exclusion gives !n / n! = 1 -1 +1/2! -1/3! +...
- Generating function for derangements is e^{-x}/(1-x)
- !15 ≈ 4.317631 × 10^12
- Subfactorial !n rounded to nearest integer is floor(n!/e + 0.5)
- Number of derangements !2 = 1
- In hat check problem, probability all hats returned wrong is !n / n! →1/e
- Number of involution derangements for n=4 is 2
- !20 ≈ 3.04140932 × 10^17
- rencontre number D(n,0) = !n, D(n,1)= C(n,1)!(n-1)
- !25 ≈ 3.106479 × 10^25
- Error in approximation |!n - n!/e| < 1/(n+1)
- !30 ≈ 2.652528598 × 10^32
Derangements Interpretation
Even as parties get bigger and the hat check becomes an absolute logistical nightmare, the chance everyone gets the wrong hat stubbornly clings to about 36.8%, proving that mathematical fate is both persistent and deeply mischievous.
Fundamental Formulas
- The number of permutations of 1 element is 1
- The number of permutations of 2 elements is 2
- The number of permutations of 3 elements is 6
- The number of permutations of 4 elements is 24
- The number of permutations of 5 elements is 120
- The number of permutations of 6 elements is 720
- The number of permutations of 7 elements is 5040
- The number of permutations of 8 elements is 40320
- The number of permutations of 9 elements is 362880
- The number of permutations of 10 elements is 3628800
- The number of permutations of 11 elements is 39916800
- The number of permutations of 12 elements is 479001600
- The number of permutations of 13 elements is 6227020800
- The number of permutations of 14 elements is 87178291200
- The number of permutations of 15 elements is 1307674368000
- The number of permutations of 16 elements is 20922789888000
- The number of permutations of 17 elements is 355687428096000
- The number of permutations of 18 elements is 6402373705728000
- The number of permutations of 19 elements is 121645100408832000
- The number of permutations of 20 elements is 2432902008176640000
- The number of permutations of 21 elements is 51090942171709440000
- The number of permutations of 22 elements is 1124000727777607680000
- The number of permutations of 23 elements is 25852016738884976640000
- The number of permutations of 24 elements is 620448401733239439360000
- The number of permutations of 25 elements is 15511210043330985984000000
- The number of permutations of 26 elements is 403291461126605635584000000
- The number of permutations of 27 elements is 10888869450418352160768000000
- The number of permutations of 28 elements is 304888344611713860501504000000
- The number of permutations of 29 elements is 8841761993739701954543616000000
- The number of permutations of 30 elements is 265252859812191058636308480000000
Fundamental Formulas Interpretation
As the number of elements grows, the number of ways to arrange them explodes with such alarming and relentless efficiency that by the time you reach 30, the resulting number is a testament to both the awesome power and the terrifying logistical nightmare of a simple factorial.
Permutation Groups
- The symmetric group S_3 has 6 elements, all permutations of 3 objects
- The order of the symmetric group S_n is n! for any n
- S_4 has 24 elements, isomorphic to rotations of tetrahedron plus reflections
- The alternating group A_n is the subgroup of even permutations, index 2 in S_n for n>1
- Number of even permutations in S_n is n!/2 for n ≥ 2
- S_5 has 120 elements, first non-solvable symmetric group
- The conjugacy classes in S_n are determined by cycle type, number equals partition number p(n)
- Center of S_n is trivial for n ≥ 3
- Derived subgroup of S_n is A_n for n ≥ 3
- S_n acts transitively on n points, sharply (n-1)! transitive
- Number of Sylow p-subgroups in S_n is n! / (p^k * m) where appropriate
- S_6 has outer automorphisms, unlike other S_n
- Holomorph of S_n includes S_n as normal subgroup
- S_n embeds into S_m for m > n via induced action
- Abelianization of S_n is Z/2Z for n ≥ 3
- S_n has presentation <x,y | x^n=y^2=(xy)^3=1> for n=3, but more complex generally
- Number of subgroups of S_4 is 30
- S_n is complete for n ≠ 6
- Schur multiplier of S_n is Z/2Z for n ≥ 4
- Cohomology ring of S_n computed via Steenrod algebra
- Representation theory: irreducible reps of S_n indexed by partitions of n
- Dimension of Specht module S^λ for partition λ
- S_n has (n choose 2) transpositions as generators
- Diameter of Cayley graph of S_n with transpositions is floor(n(n-1)/2 / log n), approx
- S_7 has 5040 elements
- Number of homomorphisms from S_3 to S_4 is 24
- S_n / A_n ≅ Z/2Z for n>1
- Number of 3-cycles in S_n is n(n-1)(n-2)/3 * 2, wait n(n-1)(n-2)
- The number of k-cycles in S_n is C(n,k) * (k-1)!
- S_infinity is the union of all S_n, locally finite
Permutation Groups Interpretation
Despite its infinite ambition, the symmetric group only truly becomes interesting when it stops being polite and starts causing trouble, which happens right around the 120th element when it declares itself unsolvable, yet remains utterly obsessed with organizing its members into neat cycles and partitions as if that were any consolation.
Sources & References
- Reference 1ENen.wikipedia.orgVisit source
- Reference 2MATHWORLDmathworld.wolfram.comVisit source
- Reference 3BRITANNICAbritannica.comVisit source
- Reference 4WOLFRAMALPHAwolframalpha.comVisit source
- Reference 5MATHmath.stackexchange.comVisit source
- Reference 6MATHSISFUNmathsisfun.comVisit source
- Reference 7BRILLIANTbrilliant.orgVisit source
- Reference 8OEISoeis.orgVisit source
- Reference 9GROUPPROPSgroupprops.subwiki.orgVisit source
- Reference 10ARXIVarxiv.orgVisit source
- Reference 11PROBABILITYCOURSEprobabilitycourse.comVisit source
- Reference 12ENen.cppreference.comVisit source