A079753 -id:A079753 - cp's OEIS frontend

A079750 Operation count to create all permutations of n distinct elements using the "streamlined" version of Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of comparisons required to find j in step L2.2'.

0, 4, 25, 156, 1099, 8800, 79209, 792100, 8713111, 104557344, 1359245485, 19029436804, 285441552075, 4567064833216, 77640102164689, 1397521838964420, 26552914940323999, 531058298806480000, 11152224274936080021

Offset: 3

Hugo Pfoertner, Jan 14 2003

The asymptotic value for large n is 0.21828...*n! See also comment for A079884

See under A079884

Hugo Pfoertner, FORTRAN program for lexicographic permutation generation.

Cf. A079884, A079751, A079752, A079753, A079754, A079755, A079756.

Fortran
```
c Program available at link.
```

a:=n->sum((n+1)!/j!,j=3..n): seq(a(n), n=2..20); # Zerinvary Lajos, Oct 20 2006

a[3] = 0; a[n_] := n*a[n - 1] + n; Table[a[n], {n, 3, 21}]

a(3)=0, a(n) = n * a(n-1) + n for n >= 4.
a(n) = Sum_{j=3..n} (n+1)!/j!. - Zerinvary Lajos, Oct 20 2006
For n >= 3, a(n) = floor((e - 5/2)*n! - 1/2). - Benoit Cloitre, Aug 03 2007

Edited and extended by Robert G. Wilson v, Jan 22 2003

A079884 Number of comparisons required to create all permutations of n distinct elements using the "streamlined" version of Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2.

Original entry on oeis.org

11, 54, 285, 1731, 12145, 97196, 874809, 8748145, 96229661, 1154756010, 15011828221, 210165595199, 3152483928105, 50439742849816, 857475628447025, 15434561312046621, 293256664928885989, 5865133298577719990, 123167799270132120021, 2709691583942906640715, 62322906430686852736721

Offset: 3

Hugo Pfoertner, Jan 12 2003

The method generates all permutations in lexicographic order. It is described in the answer to Exercise 1, Section 7.2.1.2 of Knuth's The Art of Computer Programming Vol. 4. The description is based on the Algol procedure NEXTPERM by J.P.N.Phillips. The operation counts were determined with a FORTRAN subroutine LPG. To create all permutations of n distinct elements the number of comparisons between the array elements approaches 2.410756*n! for large n (e.g. n>8).

			The "streamlined" permutation algorithm L' needs fewer comparisons a(n) than the original Algorithm L, for which the number of required comparisons between the elements to be permuted is given by A038156(n) for step L2 and A038155(n) for step L3. A038156(3)+A038155(3)=9+6=15 > a(3)=11 A038156(4)+A038155(4)=40+30=70 > a(4)=54 A038156(10)+A038155(10)=6235300+4932045=11167345 > a(10)=8748145.

D. E. Knuth: The Art of Computer Programming, Volume 4, Combinatorial Algorithms, Volume 4A, Enumeration and Backtracking. Pre-fascicle 2B, A draft of section 7.2.1.2: Generating all permutations.
J. P. N. Phillips: "Algorithm 28, PERMUTATIONS OF THE ELEMENTS OF A VECTOR IN LEXICOGRAPHIC ORDER" The Computer Journal, Volume 10, Issue 3: November 1967. (Algorithms supplement), page 311. See link.

Hugo Pfoertner, FORTRAN program for lexicographic permutation generation.
J. P. N. Phillips, Algorithm 28 from Algorithms supplement.

Partial counts given in A079750, A079753.
Number of index tests: A079885.
Cf. A000217, A000142, A038155, A038156.

Fortran
```
Program available at Pfoertner link
```

a079884(nmax) = {my(a=vector(nmax)); a[3]=11; for(k=4, nmax, a[k]=k*a[k-1]+k*(k+1)/2); a[3..nmax]} \\ Hugo Pfoertner, Jun 05 2024

a(3) = 11; a(n) = n*a(n-1) + n*(n+1)/2.
a(n) = 2*n! - 1 + A079750(n) + A079753(n).
For n>=3, a(n)=floor(c*n!-(n-3)/2) where c = lim_{n->oo} a(n)/n! = 2.4107560760219... - Benoit Cloitre; c=3*e/2-5/3 - Guido Dhondt (dhondt(AT)t-online.de), Jan 20 2003

A079756 Operation count to create all permutations of n distinct elements using the "streamlined" version of Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of interchanges in reversal step.

Original entry on oeis.org

0, 0, 4, 29, 215, 1734, 15630, 156327, 1719637, 20635688, 268264004, 3755696121, 56335441899, 901367070474, 15323240198170, 275818323567179, 5240548147776545, 104810962955531052, 2201030222066152272

Offset: 3

Hugo Pfoertner, Jan 16 2003

nonn

The asymptotic value for large n is 0.04308...*n! = (e+1/e-3)/2 * n! See also comment for A079884.

See under A079884

Indranil Ghosh, Table of n, a(n) for n = 3..101
Hugo Pfoertner, FORTRAN program for lexicographic permutation generation

Cf. A079884, A079750, A079751, A079752, A079753, A079754, A079755.

a[3] = 0; a[4] = 0; a[n_] := n*a[n - 1] + (n - 1)*(Floor[(n - 1)/2] - 1); Table[a[n ], {n, 3, 21}]

l=[0, 0, 0, 0, 0]
for n in range(5, 22):
    l.append(n*l[n - 1] + (n - 1)*((n - 1)//2 - 1))
print(l[3:]) # Indranil Ghosh, Jul 18 2017

a(3)=0, a(4)=0, a(n) = n*a(n-1) + (n-1)*(floor((n-1)/2)-1) for n>=5.
For n>=3, a(n) = floor(c*n!-(n-3)/2) where c = lim_{n->infinity} a(n)/n! = 0.04308063481524377... - Benoit Cloitre, Jan 19 2003
Recurrence: (n-5)*(n-3)*(n-2)*a(n) = (n-3)*(n^3 - 7*n^2 + 11*n - 1)*a(n-1) - (n-1)*(2*n - 5)*a(n-2) - (n-4)*(n-2)^2*(n-1)*a(n-3). - Vaclav Kotesovec, Mar 16 2014

More terms from Benoit Cloitre and Robert G. Wilson v, Jan 19 2003

A079751 Operation count to create all permutations of n distinct elements using the "streamlined" version of Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of cases where the j search loop runs beyond j=n-3.

Original entry on oeis.org

0, 1, 6, 37, 260, 2081, 18730, 187301, 2060312, 24723745, 321408686, 4499721605, 67495824076, 1079933185217, 18358864148690, 330459554676421, 6278731538852000, 125574630777040001, 2637067246317840022, 58015479418992480485, 1334356026636827051156

Offset: 3

Hugo Pfoertner, Jan 14 2003

The asymptotic value for large n is 0.051615...*n! = (e - 8/3)*n!. See also comment for A079884.

See under A079884.

Hugo Pfoertner, FORTRAN program for lexicographic permutation generation.

Cf. A079885, A079750, A079752, A079753, A079754, A079755, A079756.

a:=n->sum((n-j)!*binomial(n,j),j=4..n): seq(a(n), n=3..25); # Zerinvary Lajos, Jul 31 2006

a[3] = 0; a[n_] := n*a[n - 1] + 1; Table[a[n], {n, 3, 21}]

a(3)=0, a(n) = n * a(n-1) + 1 for n >= 4.
For n >= 3, a(n) = floor(c*n!) where c = lim_{n->infinity} a(n)/n! = 0.05161516179237856869. - Benoit Cloitre
a(n) = Sum_{j=4..n} (n-j)! * binomial(n,j). - Zerinvary Lajos, Jul 31 2006
E.g.f.: (exp(x) - Sum_{k=0..3} x^k/k!) / (1 - x). - Ilya Gutkovskiy, Jun 26 2022

Edited and extended by Robert G. Wilson v, Jan 22 2003

A079755 Operation count to create all permutations of n distinct elements using the "streamlined" version of Knuth's Algorithm L (lexicographic permutation generation).

Original entry on oeis.org

0, 3, 23, 148, 1054, 8453, 76109, 761126, 8372436, 100469287, 1306100803, 18285411320, 274281169898, 4388498718473, 74604478214169, 1342880607855178, 25514731549248544, 510294630984971051, 10716187250684392271, 235756119515056630172, 5422390748846302494198, 130137377972311259861005

Offset: 3

Hugo Pfoertner, Jan 16 2003

nonn

Sequence gives number of loop repetitions in reversal step.

The asymptotic value for large n is 0.20975...*n! = (e + 1/e - 8/3)/2 * n!. See also comment for A079884.

Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2.
See also under A079884.

Hugo Pfoertner, FORTRAN program for lexicographic permutation generation

Cf. A079884, A079750, A079751, A079752, A079753, A079754, A079756, A079885.

Fortran
```
Program available at Pfoertner link.
```

a[3] = 0; a[n_] := n*a[n - 1] + (n - 1)*Floor[(n - 1)/2]; Table[a[n], {n, 3, 21}]
RecurrenceTable[{a[3]==0,a[n]==n*a[n-1]+(n-1)Floor[(n-1)/2]},a,{n,20}] (* Harvey P. Dale, May 31 2019 *)

a(3) = 0, a(n) = n * a(n - 1) + (n - 1)*floor((n - 1)/2) for n >= 4.

a(n) = floor(c*n! - (n - 1)/2) for n > 4, where c = lim n -> infinity a(n)/n! = 0.209747301481910445... - Benoit Cloitre, Jan 19 2003

More terms from Benoit Cloitre and Robert G. Wilson v, Jan 19 2003

A079752 Operation count to create all permutations of n distinct elements using the "streamlined" version of Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of times the search for an element exchangeable with a_j has to be started.

Original entry on oeis.org

0, 2, 13, 82, 579, 4638, 41749, 417498, 4592487, 55109854, 716428113, 10029993594, 150449903923, 2407198462782, 40922373867309, 736602729611578, 13995451862619999, 279909037252399998, 5878089782300399977

Offset: 3

Hugo Pfoertner, Jan 14 2003

The asymptotic value for large n is 0.11505...*n! = (17/6-e)*n!. See also comment for A079884

Hugo Pfoertner, FORTRAN program for lexicographic permutation generation.

Cf. A079884, A079750, A079751, A079753, A079754, A079755, A079756.

a[3] = 0; a[n_] := n*a[n - 1] + n - 2; Table[a[n], {n, 3, 21}]

a(3)=0, a(n) = n * a(n-1) + n - 2 for n>=4

Edited and extended by Robert G. Wilson v, Jan 22 2003

A079754 Operation count to create all permutations of n distinct elements using the "streamlined" version of Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of times l has to be repeatedly decreased in step L3.1'.

Original entry on oeis.org

0, 1, 8, 54, 388, 3119, 28092, 280948, 3090464, 37085613, 482113024, 6749582402, 101243736108, 1619899777819, 27538296223028, 495689332014624, 9418097308277992, 188361946165559993, 3955600869476760024

Offset: 3

Hugo Pfoertner, Jan 16 2003

nonn

The asymptotic value for large n is 0.07742...*n! See also comment for A079884.

Lim_{n->infinity} a(n)/n! = 3*e/2 - 4. - Hugo Pfoertner, Sep 02 2017

See under A079884

Hugo Pfoertner, FORTRAN program for lexicographic permutation generation

Cf. A079884, A079750, A079751, A079752, A079753, A079755, A079756, A196533.

a[3] = 0; a[n_] := n*a[n - 1] + (n - 2)*(n - 3)/2; Table[a[n], {n, 3, 21}]

a(3)=0, a(n) = n*a(n-1) + (n-2)*(n-3)/2 for n>=4 a(n) = A079753(n) - A079752(n)

For n>=3 a(n)=floor(c*n!-(n-3)/2) where c=limit n --> infinity a(n)/n!=0.077422742688567853... - Benoit Cloitre, Jan 20 2003

Edited and extended by Robert G. Wilson v, Jan 22 2003

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A079884 Number of comparisons required to create all permutations of n distinct elements using the "streamlined" version of Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2.

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A079756 Operation count to create all permutations of n distinct elements using the "streamlined" version of Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of interchanges in reversal step.

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A079755 Operation count to create all permutations of n distinct elements using the "streamlined" version of Knuth's Algorithm L (lexicographic permutation generation).

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