A079755 -id:A079755 - 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

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

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

A080048 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 loop repetitions in reversal step.

Original entry on oeis.org

1, 7, 34, 182, 1107, 7773, 62212, 559948, 5599525, 61594835, 739138086, 9608795202, 134523132919, 2017846993897, 32285551902472, 548854382342168, 9879378882159177, 187708198761024543, 3754163975220491050

Offset: 2

Hugo Pfoertner, Jan 24 2003

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. Available online; see link.

Cf. A038155, A038156, A056542, A080047, A080049, A079755.

Fortran
```
! Program available at link.
```

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

c = limit n --> infinity a(n)/n! = 1.54308063481524377826 = (e+1/e)/2, a(n) = floor [c*n!-(n+1)/2] for n>=2.

A079885 Number of index tests 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

0, 4, 29, 185, 1314, 10534, 94839, 948427, 10432748, 125193032, 1627509489, 22785132925

Offset: 3

Hugo Pfoertner, Jan 13 2003

nonn

The required number of index tests (test for termination and test in the final reversion loop) becomes 0.2613625*n! for large n, if the test for n=3 is excluded. If n=3 is included the additionally required termination test adds n!/6 index comparisons, increasing the number of index comparisons to 0.428029*n! (63.8% more index comparisons).

The corresponding number of index tests needed by the "pure" Algorithm L is given by A038156(n)+A080048(n), which is 12.478..*a(n) for large n.

For references and corresponding links see under A079884

Hugo Pfoertner, FORTRAN program for lexicographic permutation generation.

Cf. A000142, partial counts given in A079751, A079755. Number of element comparisons: A079884.

Cf. A038156, A080048.

Fortran
```
! program available at link
```

a(3)=0, a(n)=n*a(n-1)+1+(n-1)*floor((n-1)/2) for n>=4 a(n) = A079751(n) + A079755(n)
For n>=3 a(n)=floor(c*n!-(n-3)/2) where c=limit n-->infinity a(n)/n!= 0.261362463274289013838... - Benoit Cloitre, Jan 20 2003
In closed form, c = 3*exp(1)/2 + exp(-1)/2 - 4. - Vaclav Kotesovec, Mar 18 2014

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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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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'.

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A080048 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 loop repetitions in reversal step.

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