A060117 -id:A060117 - cp's OEIS frontend

A261217 A(i,j) = rank (in A060118) of the composition of the i-th and the j-th permutation in table A060118, which lists all finite permutations.

0, 1, 1, 2, 0, 2, 3, 3, 5, 3, 4, 2, 0, 4, 4, 5, 5, 4, 1, 3, 5, 6, 4, 3, 5, 5, 2, 6, 7, 7, 1, 2, 1, 4, 7, 7, 8, 6, 8, 0, 0, 0, 14, 6, 8, 9, 9, 11, 9, 2, 1, 15, 15, 11, 9, 10, 8, 6, 10, 10, 3, 22, 14, 12, 10, 10, 11, 11, 10, 7, 9, 11, 23, 23, 16, 13, 9, 11, 12, 10, 9, 11, 11, 8, 0, 22, 21, 17, 17, 8, 12, 13, 13, 7, 8, 7, 10, 1, 1, 19, 20, 13, 16, 19, 13, 14, 12, 14, 6, 6, 6, 12, 0, 2, 18, 18, 12, 8, 18, 14

Offset: 0

Antti Karttunen, Aug 26 2015

The square array A(row>=0, col>=0) is read by downwards antidiagonals as: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), A(0,3), A(1,2), A(2,1), A(3,0), ...
A(i,j) gives the rank (in ordering used by table A060118) of the permutation which is obtained by composing permutations p and q listed as the i-th and the j-th permutation in irregular table A060118 (note that the identity permutation is the 0th). Here the convention is that "permutations act of the left", thus, if p1 and p2 are permutations, then the product of p1 and p2 (p1 * p2) is defined such that (p1 * p2)(i) = p1(p2(i)) for i=1...
Equally, A(i,j) gives the rank in A060117 of the composition of the i-th and the j-th permutation in A060117, when convention is that "permutations act on the right".
Each row and column is a permutation of A001477, because this is the Cayley table ("multiplication table") of an infinite enumerable group, namely, that subgroup of the infinite symmetric group (S_inf) which consists of permutations moving only finite number of elements.

			The top left corner of the array:
   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, ...
   1,  0,  3,  2,  5,  4,  7,  6,  9,  8, 11, 10, 13, ...
   2,  5,  0,  4,  3,  1,  8, 11,  6, 10,  9,  7, 14, ...
   3,  4,  1,  5,  2,  0,  9, 10,  7, 11,  8,  6, 15, ...
   4,  3,  5,  1,  0,  2, 10,  9, 11,  7,  6,  8, 16, ...
   5,  2,  4,  0,  1,  3, 11,  8, 10,  6,  7,  9, 17, ...
   6,  7, 14, 15, 22, 23,  0,  1, 12, 13, 18, 19,  8, ...
   7,  6, 15, 14, 23, 22,  1,  0, 13, 12, 19, 18,  9, ...
   8, 11, 12, 16, 21, 19,  2,  5, 14, 17, 20, 23,  6, ...
   9, 10, 13, 17, 20, 18,  3,  4, 15, 16, 21, 22,  7, ...
  10,  9, 17, 13, 18, 20,  4,  3, 16, 15, 22, 21, 11, ...
  11,  8, 16, 12, 19, 21,  5,  2, 17, 14, 23, 20, 10, ...
  12, 19,  8, 21, 16, 11, 14, 23,  2, 20, 17,  5,  0, ...
  ...
For A(1,2) (row=1, column=2, both starting from zero), we take as permutation p the permutation which has rank=1 in the ordering used by A060118, which is a simple transposition (1 2), which we can extend with fixed terms as far as we wish (e.g., like {2,1,3,4,5,...}), and as permutation q we take the permutation which has rank=2 (in the same list), which is {1,3,2}. We compose these from the left, so that the latter one, q, acts first, thus c(i) = p(q(i)), and the result is permutation {2,3,1}, which is listed as the 3rd one in A060118, thus A(1,2) = 3.
For A(2,1) we compose those two permutations in opposite order, as d(i) = q(p(i)), which gives permutation {3,1,2} which is listed as the 5th one in A060118, thus A(2,1) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..7259; the first 120 antidiagonals of array
Wikipedia, Cayley table

Transpose: A261216.
Row 0 & Column 0: A001477 (identity permutation)
Row 1: A004442.
Column 1: A261218.
Main diagonal: A261219.
Cf. also A060117, A060118, A261096, A261097.
Cf. also A089839.
Permutations used in conjugation-formulas: A060119, A060120, A060125, A060126, A060127.

By conjugating with related permutations and arrays:
A(i,j) = A060125(A261216(A060125(i),A060125(j))).
A(i,j) = A060127(A261096(A060120(i),A060120(j))).
A(i,j) = A060126(A261097(A060119(i),A060119(j))).

A178476 Permutations of 123456: Numbers having each of the decimal digits 1,...,6 exactly once, and no other digit.

Original entry on oeis.org

123456, 123465, 123546, 123564, 123645, 123654, 124356, 124365, 124536, 124563, 124635, 124653, 125346, 125364, 125436, 125463, 125634, 125643, 126345, 126354, 126435, 126453, 126534, 126543, 132456, 132465, 132546, 132564, 132645, 132654

Offset: 1

M. F. Hasler, May 28 2010

This finite sequence contains 6!=720 terms.
This is a subsequence of A030299, consisting of elements A030299(154)..A030299(873).
If individual digits are be split up into separate terms, we get a subsequence of A030298.
It would be interesting to conceive simple and/or efficient functions which yield (a) the n-th term of this sequence: f(n)=a(n), (b) for a given term, the subsequent one: f(a(n)) = a(1 + (n mod 6!)).
The expression a(n+6) - a(n) takes only 18 different values for n = 1..6!-6.
An efficient procedure for generating the n-th term of this sequence can be found at A178475. - Nathaniel Johnston, May 19 2011
From Hieronymus Fischer, Feb 13 2013: (Start)
The sum of all terms as decimal numbers is 279999720.
General formula for the sum of all terms (interpreted as decimal permutational numbers with exactly d different digits from the range 1..d < 10): sum = (d+1)!*(10^d-1)/18.
If the terms are interpreted as base-7 numbers the sum is 49412160.
General formula for the sum of all terms of the corresponding sequence of base-p permutational numbers (numbers with exactly p-1 different digits excluding the zero digit): sum = (p-2)!*(p^p-p)/2. (End)

Nathaniel Johnston, Table of n, a(n) for n = 1..720 (full sequence)

Cf. A030298, A030299, A055089, A060117.

Take[FromDigits/@Permutations[Range[6]],40] (* Harvey P. Dale, Jun 05 2012 *)

v=vector(6,i,10^(i-1))~; A178476=vecsort(vector(6!,i,numtoperm(6,i)*v));
is_A178476(x)= { vecsort(Vec(Str(x)))==Vec("123456") }
forstep( m=123456,654321,9, is_A178476(m) & print1(m","))

a(n) + a(6! + 1 - n) = 777777.
floor( a(n) / 10^5 ) = ceiling( n / 5! ).
a(n) = A030299(n+153).
a(n) == 3 (mod 9).
a(n) = 3 + 9*A178486(n).

A178485 (A178475(n)-6)/9.

Original entry on oeis.org

1371, 1372, 1381, 1383, 1392, 1393, 1471, 1472, 1491, 1494, 1502, 1504, 1581, 1583, 1591, 1594, 1613, 1614, 1692, 1693, 1702, 1704, 1713, 1714, 2371, 2372, 2381, 2383, 2392, 2393, 2571, 2572, 2601, 2605, 2612, 2615, 2681, 2683, 2701, 2705, 2723, 2725

Offset: 1

M. F. Hasler, May 28 2010

There are 5!=120 terms in this finite sequence. Its origin is the fact that numbers whose decimal expansion is a permutation of 12345 are all of the form 9k+6.

Nathaniel Johnston, Table of n, a(n) for n = 1..120 (full sequence)

Cf. A030298, A030299, A055089, A060117, A178486, A191819, A191820.

v=vector(5,i,10^(i-1))~; vecsort(vector(5!,i,numtoperm(5,i)*v))
is_A178475(x)= { vecsort(Vec(Str(x)))==Vec("12345") }
forstep( m=12345,54321,9, is_A178475(m) & print1(m","))

a(n) + a(5!+1-n) = 7406.
a(n) == 1, 2, 3, 4 or 5 (mod 10).
a(n+6)-a(n) is an element of { 100, 110, 111, 200, 220, 222, 679 }.
a(n+6)-a(n) = 679 iff (n-1)%24 > 17, where % denotes the remainder upon division.
a(n+6)-a(n) = 200, 220 or 222 iff (n-1)%30 > 23, i.e. n==25,...,30 (mod 30).

A178486 (A178476(n)-3)/9.

Original entry on oeis.org

13717, 13718, 13727, 13729, 13738, 13739, 13817, 13818, 13837, 13840, 13848, 13850, 13927, 13929, 13937, 13940, 13959, 13960, 14038, 14039, 14048, 14050, 14059, 14060, 14717, 14718, 14727, 14729, 14738, 14739, 14917, 14918, 14947, 14951, 14958, 14961

Offset: 1

M. F. Hasler, May 28 2010

The sequence is motivated by the fact that numbers whose decimal expansion is a permutation of 123456, are all of the form 9k+3.

There are 6!=720 terms in this finite sequence.

Nathaniel Johnston, Table of n, a(n) for n = 1..720 (full sequence)

Cf. A030298, A030299, A055089, A060117, A178485, A191819, A191820.

forstep( m=123456,654321/*or less*/,9, is_A178476(m) & print1(m\9",")) /*cf. A178476*/

a(n) + a(6!+1-n) = 86419.

a(n) == 0, 1, 2, 7, 8 or 9 (mod 10).

A220658 Irregular table, where the n-th row consists of A084558(n)+1 copies of n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20

Offset: 0

Antti Karttunen, Dec 18 2012

Equally, for n>=1, each i in range [n!,(n+1)!-1] occurs n+1 times.

Used for computing A220659, A055089 and A060118: The n-th term a(n) tells which permutation (counted from the start, zero-based) of A055089 or A060117/A060118 the n-th term in those sequence belongs to.

			Rows of this irregular table begin as:
0;
1, 1;
2, 2, 2;
3, 3, 3;
4, 4, 4;
5, 5, 5;
6, 6, 6, 6;
The terms A055089(3), A055089(4) and A055089(5) are 1,3,2. As a(3), a(4) and a(5) are all 2, we see that "132" is the second permutation in A055089-list, after the identity permutation "1", which has the index zero.

A. Karttunen, Rows 0..720 of the irregular table, flattened.

Cf. A220657, A220659.

A220659 Irregular table: row n (n >= 1) consists of numbers 0..A084558(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3

Offset: 0

Antti Karttunen, Dec 18 2012

The term a(n) gives the position (zero-based, starting from the left hand end of permutation) of a corresponding permutation A055089 and A060117/A060118 from which the term A055089(n) or A060118(n) is to be picked.

			Rows of this irregular table begin as:
0;
0, 1;
0, 1, 2;
0, 1, 2;
0, 1, 2;
0, 1, 2;
0, 1, 2, 3;
The term 2 occurs four times in A084558, in positions 2, 3, 4 and 5. Thus rows 2, 3, 4 and 5 (zero-based) of this irregular table are all 0,1,2.

Antti Karttunen, Rows 0..720 of the irregular table, flattened.

(define (A220659 n) (- n (A220657 (A220658 n))))

a(n) = n - A220657(A220658(n)).

A060500 a(n) = number of drops in the n-th permutation of list A060118; the average of digits (where "digits" may eventually obtain also any values > 9) in each siteswap pattern A060496(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 1, 1, 1, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 1, 2, 2, 2, 2, 3, 2, 3, 2

Offset: 0

Antti Karttunen, Mar 22 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A060125, A060129, A060501, A060502, A275849, A275851.

A060500 := avg(Perm2SiteSwap1(PermUnrank3R(n)));
# PermUnrank3R(r) gives the permutation with rank r in list A060117:
PermUnrank3R := proc(r) local n; n := nops(factorial_base(r)); convert(PermUnrank3Raux(n+1, r, []), 'permlist', 1+(((r+2) mod (r+1))*n)); end;
PermUnrank3Raux := proc(n, r, p) local s; if(0 = r) then RETURN(p); else s := floor(r/((n-1)!)); RETURN(PermUnrank3Raux(n-1, r-(s*((n-1)!)), permul(p, [[n, n-s]]))); fi; end;
Perm2SiteSwap1 := proc(p) local ip, n, i, a; n := nops(p); ip := convert(invperm(convert(p, 'disjcyc')), 'permlist', n); a := []; for i from 1 to n do a := [op(a), ((ip[i]-i) mod n)]; od; RETURN(a); end;
avg := a -> (convert(a,`+`)/nops(a));

(define (A060500 n) (let ((s (+ 1 (A084558 n))) (p (A060118permvec-short n))) (let loop ((d 0) (i 1)) (if (> i s) d (loop (+ d (if (< (vector-ref p (- i 1)) i) 1 0)) (+ 1 i))))))
(define (A060118permvec-short rank) (permute-A060118 (make-initialized-vector (+ 1 (A084558 rank)) 1+) (+ 1 (A084558 rank)) rank))
(define (permute-A060118 elems size permrank) (let ((p (vector-head elems size))) (let unrankA060118 ((r permrank) (i 1)) (cond ((zero? r) p) (else (let* ((j (1+ i)) (m (modulo r j))) (cond ((not (zero? m)) (let ((org-i (vector-ref p i))) (vector-set! p i (vector-ref p (- i m))) (vector-set! p (- i m) org-i)))) (unrankA060118 (/ (- r m) j) j)))))))

From Antti Karttunen, Aug 18 2016: (Start)
The following formula reflects the original definition of computing the average, with a few unnecessary steps eliminated:
a(n) = 1/s * Sum_{i=1..s} ((i-p[i]) modulo s), where p is the permutation of rank n as ordered in the list A060117, and s is its size (the number of its elements) computed as s = 1+A084558(n).
a(n) = 1/s * Sum_{i=1..s} ((p[i]-i) modulo s). [If inverse permutations from list A060118 are used, then we just flip the order of difference that is used in the first formula].
a(n) = Sum_{i=1..s} [p[i]A060502 for the proof].
a(n) = A060502(A060125(n)).
a(n) = A060129(n) - A060502(n).
a(n) = A060501(n) - A275851(n) = 1 + A275849(n) - A275851(n).
(End)

Maple code collected together, alternative definition and new formulas added by Antti Karttunen, Aug 24 2016

A178477 Permutations of 1234567: Numbers having each of the decimal digits 1,...,7 exactly once, and no other digit.

Original entry on oeis.org

1234567, 1234576, 1234657, 1234675, 1234756, 1234765, 1235467, 1235476, 1235647, 1235674, 1235746, 1235764, 1236457, 1236475, 1236547, 1236574, 1236745, 1236754, 1237456, 1237465, 1237546, 1237564, 1237645, 1237654

Offset: 1

M. F. Hasler, Oct 09 2010

It would be nice to have a simple explicit formula for the n-th term.
Contains A000142(7) = 5040 terms. - R. J. Mathar, Apr 08 2011
An efficient procedure for generating the n-th term of this sequence can be found at A178475. - Nathaniel Johnston, May 19 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..5040 (full sequence)

Cf. A030298, A030299, A055089, A060117, A178475, A178476.

FromDigits/@Take[Permutations[Range[7]],50] (* Harvey P. Dale, Nov 11 2012 *)

is_A178477(x)= { vecsort(Vec(Str(x)))==Vec("1234567") }

A178478 Permutations of 12345678: Numbers having each of the decimal digits 1..8 exactly once, and no other digit.

Original entry on oeis.org

12345678, 12345687, 12345768, 12345786, 12345867, 12345876, 12346578, 12346587, 12346758, 12346785, 12346857, 12346875, 12347568, 12347586, 12347658, 12347685, 12347856, 12347865, 12348567, 12348576, 12348657, 12348675, 12348756, 12348765

Offset: 1

M. F. Hasler, Oct 09 2010

It would be nice to have a simple explicit formula for the n-th term.

An efficient procedure for generating the n-th term of this sequence can be found at A178475. - Nathaniel Johnston, May 19 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A030298, A030299, A055089, A060117, A178475-A178477, A191820.

Take[FromDigits/@Permutations[Range[8]],40] (* Harvey P. Dale, Oct 29 2014 *)

is_A178478(x)= { vecsort(Vec(Str(x)))==Vec("12345678") }

A178478(n)={my(b=vector(7,k,1+(n-1)%(k+1)!\k!),t=b[7], d=vector(7,i,i+(i>=t)));for(i=1,6,t=10*t+d[b[7-i]]; d=vecextract(d,Str("^"b[7-i]))); t*10+d[1]} \\ - M. F. Hasler (following N. Johnston's comment), Jan 10 2012

A261218 Row 1 of A261216.

Original entry on oeis.org

1, 0, 5, 4, 3, 2, 7, 6, 11, 10, 9, 8, 19, 18, 23, 22, 21, 20, 13, 12, 17, 16, 15, 14, 25, 24, 29, 28, 27, 26, 31, 30, 35, 34, 33, 32, 43, 42, 47, 46, 45, 44, 37, 36, 41, 40, 39, 38, 49, 48, 53, 52, 51, 50, 55, 54, 59, 58, 57, 56, 67, 66, 71, 70, 69, 68, 61, 60, 65, 64, 63, 62, 97, 96, 101, 100, 99, 98, 103, 102, 107, 106, 105, 104, 115, 114, 119, 118, 117, 116, 109, 108, 113, 112, 111, 110, 73

Offset: 0

Antti Karttunen, Aug 26 2015

nonn

Equally, column 1 of A261217.
Take the n-th (n>=0) permutation from the list A060117, change 1 to 2 and 2 to 1 to get another permutation, and note its rank in the same list to obtain a(n).
Equally, we can take the n-th (n>=0) permutation from the list A060118, swap the elements in its two leftmost positions, and note the rank of that permutation in A060118 to obtain a(n).
Self-inverse permutation of nonnegative integers.

			In A060117 the permutation with rank 2 is [1,3,2], and swapping the elements 1 and 2 we get permutation [2,3,1], which is listed in A060117 as the permutation with rank 5, thus a(2) = 5.
Equally, in A060118 the permutation with rank 2 is [1,3,2], and swapping the elements in the first and the second position gives permutation [3,1,2], which is listed in A060118 as the permutation with rank 5, thus a(2) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..5039
Index entries for sequences that are permutations of the natural numbers

Row 1 of A261216, column 1 of A261217.
Cf. A060117, A060118.
Cf. also A004442.
Related permutations: A060119, A060126, A261098.

a(n) = A261216(1,n).
By conjugating related permutations:
a(n) = A060126(A261098(A060119(n))).

cp's OEIS Frontend

A261217 A(i,j) = rank (in A060118) of the composition of the i-th and the j-th permutation in table A060118, which lists all finite permutations.

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A178476 Permutations of 123456: Numbers having each of the decimal digits 1,...,6 exactly once, and no other digit.

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A178485 (A178475(n)-6)/9.

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A178486 (A178476(n)-3)/9.

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A220658 Irregular table, where the n-th row consists of A084558(n)+1 copies of n.

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A220659 Irregular table: row n (n >= 1) consists of numbers 0..A084558(n).

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A060500 a(n) = number of drops in the n-th permutation of list A060118; the average of digits (where "digits" may eventually obtain also any values > 9) in each siteswap pattern A060496(n).

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A178477 Permutations of 1234567: Numbers having each of the decimal digits 1,...,7 exactly once, and no other digit.

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