A079756 -id:A079756 - 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

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

A079753 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 total executions of step L3.1'.

Original entry on oeis.org

0, 3, 21, 136, 967, 7757, 69841, 698446, 7682951, 92195467, 1198541137, 16779575996, 251693640031, 4027098240601, 68460670090337, 1232292061626202, 23413549170897991, 468270983417959991, 9833690651777160001

Offset: 3

Hugo Pfoertner, Jan 16 2003

nonn

The asymptotic value for large n is 0.19247...*n! = (e/2-7/6)*n!. See also comment for A079884.

See under A079884

Hugo Pfoertner, FORTRAN program for lexicographic permutation generation

Cf. A079884, A079750, A079751, A079752, A079754, A079755, A079756.

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

a(3)=0, a(n)= n*a(n-1) + (n-1)*(n-2)/2 for n>=4 a(n) = A079752(n) + A079754(n)

For n>=3, a(n)=floor(c*n!-(n-1)/2) where c=limit n-->infinity a(n)/n!= 0.192474247562855951... - Benoit Cloitre, Jan 20 2003

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

A080049 Operation count to create all permutations of n distinct elements using Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of interchange operations in step L4.

Original entry on oeis.org

0, 2, 11, 63, 388, 2734, 21893, 197069, 1970726, 21678036, 260136487, 3381774403, 47344841720, 710172625898, 11362762014473, 193166954246169, 3477005176431178, 66063098352192544, 1321261967043851051, 27746501307920872271, 610423028774259190172, 14039729661807961374198

Offset: 2

Hugo Pfoertner, Jan 24 2003

Donald E. Knuth: The Art of Computer Programming, Volume 4, Fascicle 2, Generating All Tuples and Permutations. Addison-Wesley (2005). Chapter 7.2.1.2, 39-40.

D. E. Knuth, TAOCP Vol. 4, Pre-fascicle 2b (generating all permutations).
R. J. Ord-Smith, Generation of permutation sequences: Part 1, Computer J., 13 (1970), 151-155.
Hugo Pfoertner, FORTRAN implementation of Knuth's Algorithm L for lexicographic permutation generation.

Cf. A080047, A080048, A038155, A038156, A056542, A073743, A079756, A080047, A080048.

c FORTRAN program available at Pfoertner link.

a(2)=0, a(n)=n*a(n-1) + (n-1)*floor((n-1)/2).
c = limit n ->infinity a(n)/n! = 0.5430806.. = (e+1/e)/2-1 = A073743 - 1.
a(n) = floor (c*n! - (n-1)/2) for n>=2.

A080093 Let sum(k>=0, k^n/(2k+1)!) = (x(n)e + y(n)/e)/z(n), where x(n) and z(n) are >0, then a(n)=x(n).

Original entry on oeis.org

0, 1, 1, 2, 11, 41, 81, 715, 3425, 8861, 98253, 580317, 1816640, 24011157, 166888165, 608035190, 9264071767, 73600798037, 304238004061, 5224266196935, 46499892038437, 214184962059157, 4078345814329009, 40073660040755337

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 28 2003

nonn

			Values of sum(k>=0,k^n/(2*k+1)!) = (x(n)*e + y(n)/e)/z(n) are given by n=1: (1/e)/2 = 0.183939720585721160..., n=2: (e - 3/e)/8 = 0.201830438118089783..., n=3: (e + 3/e)/16 = 0.238870009498335762..., n=4: (2e - 1/e)/16 = 0.316792763484165509..., n=5: (11e + 3/e)/64 = 0.484449038071309758..., n=6: (41e - 5/e)/128 = 0.856329357507528461..., n=7: (81e - 2/e)/128 = 1.71441460330343577..., n=8: (715e - 5/e)/512 = 3.79244552762179713..., n=9: (3425e + 55/e)/1024 = 9.11166858568033130..., n=10: (8861e + 106/e)/1024 = 23.5602446315818092...

Cf. A080094, A080095, A079750 - A079756.

cp's OEIS Frontend

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Fortran

Maple

Mathematica

Formula

Extensions

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Fortran

Mathematica

Formula

Extensions

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A079753 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 total executions of step L3.1'.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A080049 Operation count to create all permutations of n distinct elements using Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of interchange operations in step L4.

Views

Author

Keywords

References

Links

Crossrefs

Programs

A080093 Let sum(k>=0, k^n/(2k+1)!) = (x(n)e + y(n)/e)/z(n), where x(n) and z(n) are >0, then a(n)=x(n).