A060118 -id:A060118 - cp's OEIS frontend

A275851 a(n) = number of elements in range [1..(1+A084558(n))] fixed by the permutation with rank n of permutation list A060117 (or A060118).

1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 3, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 3, 1, 2, 1, 2, 1, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 3, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 0, 2, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 3, 2, 1, 1, 2, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 4

Offset: 0

Antti Karttunen, Aug 11 2016

nonn

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

Cf. A060117, A060118.
Cf. A056169, A060129, A084558, A275725.
Cf. A275852 (indices of zeros).

a(n) = A056169(A275725(n)).

a(n) = 1 + A084558(n) - A060129(n).

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.

Original entry on oeis.org

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))).

A278225 Filter-sequence for factorial base (cycles in A060117/A060118-permutations): Least number with the same prime signature as A275725.

Original entry on oeis.org

2, 4, 12, 8, 12, 8, 60, 36, 24, 16, 24, 16, 60, 24, 24, 16, 36, 16, 60, 24, 36, 16, 24, 16, 420, 180, 180, 72, 180, 72, 120, 72, 48, 32, 48, 32, 120, 48, 48, 32, 72, 32, 120, 48, 72, 32, 48, 32, 420, 180, 120, 48, 120, 48, 120, 72, 48, 32, 48, 32, 180, 72, 48, 32, 72, 32, 180, 72, 72, 32, 48, 32, 420, 120, 120, 48, 180, 48, 180, 72, 48, 32, 72, 32, 120, 48, 48

Offset: 0

Antti Karttunen, Nov 16 2016

nonn

This sequence can be used for filtering certain sequences related to cycle-structures in finite permutations as ordered by lists A060117 / A060118 (and thus also related to factorial base representation, A007623) because it matches only with any such sequence b that can be computed as b(n) = f(A275725(n)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under the index entry for "sequences computed from exponents in ...").

Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.

Cf. A007623, A060117, A060118, A046523, A275725.
Other filter-sequences related to factorial base: A278234, A278235, A278236.
Sequences that partition N into same or coarser equivalence classes: A048764, A048765, A060129, A060130, A060131, A084558, A275803, A275851, A257510.

(define (A278225 n) (A046523 (A275725 n)))

a(n) = A046523(A275725(n)).

A275832 Size of the cycle containing element 1 in finite permutations listed in tables A060117 & A060118: a(n) = A007814(A275725(n)).

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 4, 1, 3, 1, 4, 2, 4, 2, 3, 2, 4, 3, 4, 1, 2, 1, 3, 2, 3, 1, 2, 1, 5, 4, 5, 1, 4, 1, 5, 2, 5, 3, 4, 3, 5, 4, 5, 1, 2, 1, 4, 3, 4, 1, 2, 1, 5, 4, 5, 1, 3, 1, 5, 3, 5, 2, 3, 2, 5, 4, 5, 1, 3, 1, 4, 2, 4, 1, 3, 1, 5, 3, 5, 1, 4, 1, 5, 2, 5, 2, 4, 2, 5, 3, 5, 2, 3, 2, 4, 3, 4, 2, 3, 2, 5, 4, 5, 2, 4, 2, 5, 3, 5, 3, 4, 3, 5, 4, 5, 1

Offset: 0

Antti Karttunen, Aug 11 2016

nonn

			For n=0, the permutation with rank 0 in list A060118 is "1" (identity permutation) where 1 is fixed (in a 1-cycle), thus a(0)=1.
For n=1, the permutation with rank 1 in list A060118 is "21" where 1 is in a transposition (a 2-cycle), thus a(1)=2.
For n=3, the permutation with rank 3 in list A060118 is "231" where 1 is in a 3-cycle, thus a(3)=3.
For n=16, the permutation with rank 16 in list A060118 is "3412" (1 is in the other of two disjoint transpositions (1 3) and (2 4)), thus a(16)=2.
For n=44, the permutation with rank 44 in list A060118 is "43251", where 1 is a part of 3-cycle, thus a(44)=3.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Cf. A060117, A060118.
Cf. A000142, A007814, A033312, A275725, A275807.
Cf. A153880 (positions of 1's), A273670 (of terms larger than one), A275833 (of odd terms), A275834 (of even terms).

(define (A275832 n) (A007814 (A275725 n)))

a(n) = A007814(A275725(n)).
Other identities:
For n >= 1, a(A033312(n)) = n.
For n >= 2, a(A000142(n)) = 1.

A275834 Positions of even terms in A275832; indices of those permutations in tables A060117 & A060118 where element 1 is in an even cycle.

Original entry on oeis.org

1, 4, 7, 9, 11, 15, 16, 17, 18, 20, 21, 23, 25, 28, 31, 34, 37, 40, 43, 46, 49, 51, 53, 55, 58, 66, 68, 70, 75, 76, 77, 85, 88, 90, 91, 92, 96, 98, 99, 101, 102, 104, 106, 108, 109, 110, 115, 118, 121, 124, 127, 129, 131, 135, 136, 137, 138, 140, 141, 143, 145, 148, 151, 153, 155, 159, 160, 161, 162, 164, 165, 167, 169, 172, 175, 177, 179

Offset: 1

Antti Karttunen, Aug 11 2016

nonn

Equally: positions of even terms in A275726.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Complement: A275833.
A subsequence of A273670 and A275814.
Cf. A060117, A060118, A275726, A275832.

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

A275803 a(n) = A051903(A275725(n)); maximal cycle sizes of finite permutations listed in the order A060117 / A060118.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 2, 3, 4, 3, 4, 2, 3, 3, 4, 2, 4, 2, 3, 2, 4, 3, 4, 2, 2, 2, 3, 2, 3, 3, 3, 4, 5, 4, 5, 3, 4, 4, 5, 3, 5, 3, 4, 3, 5, 4, 5, 2, 2, 3, 4, 3, 4, 3, 3, 4, 5, 4, 5, 2, 3, 4, 5, 3, 5, 2, 3, 3, 5, 4, 5, 2, 3, 3, 4, 2, 4, 2, 3, 4, 5, 3, 5, 3, 4, 4, 5, 3, 5, 2, 4, 3, 5, 3, 5, 2, 3, 2, 4, 3, 4, 2, 3, 3, 5, 4, 5, 2, 4, 3, 5, 3, 5, 3, 4, 3, 5, 4, 5, 2

Offset: 0

Antti Karttunen, Aug 10 2016

			For n=27, which in factorial base (A007623) is "1011" and encodes (in A060118-order) permutation "23154" with one 3-cycle and one 2-cycle, the longest cycle has three elements, thus a(27) = 3.

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

Cf. A007623, A060117, A060118.
Cf. A051903, A275725.
Cf. A261220 (gives the positions of 1 and 2's).
Differs from A060131 for the first time at n=27, where a(27) = 3, while A060131(27) = 6.

(define (A275803 n) (A051903 (A275725 n)))

a(n) = A051903(A275725(n)).

A275813 Positions of odd terms in A060131; indices of permutations of an odd order in tables A060117 & A060118.

Original entry on oeis.org

0, 3, 5, 8, 10, 13, 14, 19, 22, 30, 33, 35, 36, 39, 41, 42, 45, 47, 50, 52, 54, 57, 59, 63, 65, 69, 71, 73, 74, 81, 83, 84, 87, 89, 93, 95, 97, 100, 105, 107, 111, 113, 114, 117, 119, 144, 147, 149, 152, 154, 157, 158, 163, 166, 168, 171, 173, 176, 178, 181, 182, 187, 190, 192, 195, 197, 200, 202, 205, 206, 211, 214, 216, 219, 221

Offset: 0

Antti Karttunen, Aug 10 2016

nonn

Indexing starts with zero, because a(0) = 0 (indicating an identity permutation) is a special case in this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to factorial base representation

Complement: A275814.
Cf. A060117, A060118, A060131.
Cf. A275809 (a subsequence).

A275814 Positions of even terms in A060131; indices of permutations of an even order in tables A060117 & A060118.

Original entry on oeis.org

1, 2, 4, 6, 7, 9, 11, 12, 15, 16, 17, 18, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 37, 38, 40, 43, 44, 46, 48, 49, 51, 53, 55, 56, 58, 60, 61, 62, 64, 66, 67, 68, 70, 72, 75, 76, 77, 78, 79, 80, 82, 85, 86, 88, 90, 91, 92, 94, 96, 98, 99, 101, 102, 103, 104, 106, 108, 109, 110, 112, 115, 116, 118, 120, 121, 122, 123, 124, 125

Offset: 1

Antti Karttunen, Aug 10 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to factorial base representation

Complement: A275813.
Cf. A060117, A060118, A060131.
Cf. A275834 (a subsequence).

A275833 Positions of odd terms in A275832; indices of those permutations in tables A060117 & A060118 where element 1 is in an odd cycle.

Original entry on oeis.org

0, 2, 3, 5, 6, 8, 10, 12, 13, 14, 19, 22, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 52, 54, 56, 57, 59, 60, 61, 62, 63, 64, 65, 67, 69, 71, 72, 73, 74, 78, 79, 80, 81, 82, 83, 84, 86, 87, 89, 93, 94, 95, 97, 100, 103, 105, 107, 111, 112, 113, 114, 116, 117, 119, 120, 122, 123, 125, 126, 128, 130, 132, 133, 134, 139, 142, 144

Offset: 0

Antti Karttunen, Aug 11 2016

nonn

Indexing starts from zero, because a(0)=0 is a special case in this sequence.

Equally: positions of odd terms in A275726.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Complement: A275834.
Cf. A153880 (a subsequence).
Cf. A275726, A275832.

cp's OEIS Frontend

A275851 a(n) = number of elements in range [1..(1+A084558(n))] fixed by the permutation with rank n of permutation list A060117 (or A060118).

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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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A278225 Filter-sequence for factorial base (cycles in A060117/A060118-permutations): Least number with the same prime signature as A275725.

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A275832 Size of the cycle containing element 1 in finite permutations listed in tables A060117 & A060118: a(n) = A007814(A275725(n)).

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A275834 Positions of even terms in A275832; indices of those permutations in tables A060117 & A060118 where element 1 is in an even cycle.

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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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Maple

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A275803 a(n) = A051903(A275725(n)); maximal cycle sizes of finite permutations listed in the order A060117 / A060118.

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A275813 Positions of odd terms in A060131; indices of permutations of an odd order in tables A060117 & A060118.

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A275814 Positions of even terms in A060131; indices of permutations of an even order in tables A060117 & A060118.

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A275833 Positions of odd terms in A275832; indices of those permutations in tables A060117 & A060118 where element 1 is in an odd cycle.

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