A261219 -id:A261219 - cp's OEIS frontend

A060126 Positions of permutations of A055089 in the permutation sequence A060117.

0, 1, 2, 3, 5, 4, 6, 7, 8, 9, 11, 10, 14, 15, 12, 13, 16, 17, 23, 22, 19, 18, 21, 20, 24, 25, 26, 27, 29, 28, 30, 31, 32, 33, 35, 34, 38, 39, 36, 37, 40, 41, 47, 46, 43, 42, 45, 44, 54, 55, 56, 57, 59, 58, 48, 49, 50, 51, 53, 52, 60, 61, 62, 63, 65, 64, 67, 66, 71, 70, 68, 69

Offset: 0

Antti Karttunen, Mar 02 2001

Together with the inverse A060119 this can be used to conjugate between "multiplication tables" of A261096 & A261216 (and for example, their main diagonals A261099 & A261219, or between involutions A056019 & A060125, see the Formula section) that have been computed for these two common alternative orderings of permutations. - Antti Karttunen, Sep 28 2016

Inverse: A060119.
Cf. A060132 (fixed points).
Cf. A055089, A060117.
Cf. also A056019, A060125, A060127, A261096, A261099, A261216, A261219.

# Procedure PermRank3R is given in A060125 and PermRevLexUnrank in A055089:
A060126(n) = PermRank3R(PermRevLexUnrank(n));

Other identities. For all n >= 0:

a(A056019(A060119(n))) = A060125(n).

Edited by Antti Karttunen, Sep 28 2016

A060119 Positions of permutations of A060117 in reversed colexicographic ordering A055089.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 8, 9, 11, 10, 14, 15, 12, 13, 16, 17, 21, 20, 23, 22, 19, 18, 24, 25, 26, 27, 29, 28, 30, 31, 32, 33, 35, 34, 38, 39, 36, 37, 40, 41, 45, 44, 47, 46, 43, 42, 54, 55, 56, 57, 59, 58, 48, 49, 50, 51, 53, 52, 60, 61, 62, 63, 65, 64, 67, 66, 70, 71, 69, 68

Offset: 0

Antti Karttunen, Mar 02 2001

Together with the inverse A060126 this can be used to conjugate between "multiplication tables" of A261096 & A261216 (and for example, their main diagonals A261099 & A261219, or between involutions A056019 & A060125, see the Formula section) that have been computed for these two common alternative orderings of permutations. - Antti Karttunen, Sep 28 2016

Inverse: A060126.
Cf. A060132 (fixed points).
Cf. A055089, A060117.
Cf. also A056019, A060120, A060125, A261096, A261099, A261216, A261219.

# The procedure PermUnrank3R is given in A060117, and PermRevLexRank in A056019:
A060119(n) = PermRevLexRank(PermUnrank3R(n));

As a composition of other permutations:
a(n) = A056019(A060120(n)).
Other identities, for all n >= 0:
a(A060125(A060126(n))) = A056019(n).

Edited by Antti Karttunen, Sep 27 2016

A261220 Ranks of involutions in permutation orderings A060117 and A060118.

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 12, 16, 18, 20, 24, 25, 26, 28, 48, 49, 60, 66, 72, 76, 78, 90, 96, 98, 102, 108, 120, 121, 122, 124, 126, 127, 132, 136, 138, 140, 240, 241, 242, 244, 288, 289, 312, 316, 336, 338, 360, 361, 372, 378, 384, 385, 432, 450, 456, 468, 480, 484, 486, 498, 504, 508, 528, 546, 576, 582, 600, 602, 606, 612, 624, 626, 648, 660, 672, 678, 720, 721

Offset: 0

Antti Karttunen, Aug 26 2015

From Antti Karttunen, Aug 17 2016: (Start)
Intersection of A275804 and A276005. In other words, these are numbers in whose factorial base representation (A007623, see A260743) there does not exist any such pair of nonzero digits d_i and d_j in positions i and j that either (i - d_i) = j or (i - d_i) = (j - d_j) would hold. Here one-based indexing is used so that the least significant digit at right is in position 1.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..1000
Index entries for sequences related to factorial base representation

Intersection of A275804 and A276005.
Same sequence shown in factorial base: A260743.
Cf. A060117, A060118, A275947, A276007.
Positions of zeros in A261219.
Positions of 1 and 2's in A060131 and A275803.
Subsequence: A060112.
Cf. also A014489.

A261216 A(i,j) = rank (in A060117) of the composition of the i-th and the j-th permutation in table A060117, which lists all finite permutations.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 5, 3, 3, 4, 4, 0, 2, 4, 5, 3, 1, 4, 5, 5, 6, 2, 5, 5, 3, 4, 6, 7, 7, 4, 1, 2, 1, 7, 7, 8, 6, 14, 0, 0, 0, 8, 6, 8, 9, 11, 15, 15, 1, 2, 9, 11, 9, 9, 10, 10, 12, 14, 22, 3, 10, 10, 6, 8, 10, 11, 9, 13, 16, 23, 23, 11, 9, 7, 10, 11, 11, 12, 8, 17, 17, 21, 22, 0, 8, 11, 11, 9, 10, 12, 13, 19, 16, 13, 20, 19, 1, 1, 10, 7, 8, 7, 13, 13, 14, 18, 8, 12, 18, 18, 2, 0, 12, 6, 6, 6, 14, 12, 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 of the permutation (in ordering used by table A060117) which is obtained by composing permutations p and q listed as the i-th and the j-th permutation in irregular table A060117 (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 A060118 of the composition of the i-th and the j-th permutation in A060118, 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,  5,  4,  3,  2,  7,  6, 11, 10,  9,  8, 19, ...
   2,  3,  0,  1,  5,  4, 14, 15, 12, 13, 17, 16,  8, ...
   3,  2,  4,  5,  1,  0, 15, 14, 16, 17, 13, 12, 21, ...
   4,  5,  3,  2,  0,  1, 22, 23, 21, 20, 18, 19, 16, ...
   5,  4,  1,  0,  2,  3, 23, 22, 19, 18, 20, 21, 11, ...
   6,  7,  8,  9, 10, 11,  0,  1,  2,  3,  4,  5, 14, ...
   7,  6, 11, 10,  9,  8,  1,  0,  5,  4,  3,  2, 23, ...
   8,  9,  6,  7, 11, 10, 12, 13, 14, 15, 16, 17,  2, ...
   9,  8, 10, 11,  7,  6, 13, 12, 17, 16, 15, 14, 20, ...
  10, 11,  9,  8,  6,  7, 18, 19, 20, 21, 22, 23, 17, ...
  11, 10,  7,  6,  8,  9, 19, 18, 23, 22, 21, 20,  5, ...
  12, 13, 14, 15, 16, 17,  8,  9,  6,  7, 11, 10,  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 A060117, 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 5th one in A060117, thus A(1,2) = 5.
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 3rd one in A060117, thus A(2,1) = 3.

Antti Karttunen, Table of n, a(n) for n = 0..10439; the first 144 antidiagonals of array
Wikipedia, Cayley table

Transpose: A261217.
Row 0 & Column 0: A001477 (identity permutation).
Row 1: A261218.
Column 1: A004442.
Main diagonal: A261219.
Cf. also A060117, A060118, A261096, A261097.
Permutations used in conjugation-formulas: A060119, A060120, A060125, A060126, A060127.

By conjugating with related permutations and arrays:
A(i,j) = A060125(A261217(A060125(i),A060125(j))).
A(i,j) = A060126(A261096(A060119(i),A060119(j))).
A(i,j) = A060127(A261097(A060120(i),A060120(j))).

A261099 Main diagonal of A261096.

Original entry on oeis.org

0, 0, 0, 4, 3, 0, 0, 0, 12, 16, 23, 19, 8, 23, 0, 20, 0, 7, 16, 11, 15, 0, 7, 0, 0, 0, 0, 4, 3, 0, 48, 48, 60, 64, 71, 67, 86, 93, 74, 94, 74, 85, 116, 111, 119, 99, 108, 99, 30, 30, 86, 89, 112, 111, 0, 0, 78, 82, 107, 103, 0, 20, 26, 46, 96, 103, 15, 0, 41, 29, 78, 73, 60, 115, 38, 119, 38, 63, 56, 107, 0, 104, 0, 55, 26, 100, 0, 104, 19, 42, 33, 56, 11, 52, 0, 25

Offset: 0

Antti Karttunen, Aug 26 2015

nonn

Equally: main diagonal of A261097.

For permutation p, which has rank n in permutation list A055089 (A195663), a(n) gives the rank of the "square" of that permutation (obtained by composing it with itself as: q(i) = p(p(i))) in the same list. Thus zeros (which mark the identity permutation, with rank 0) occur at positions where the permutations of A055089/A195663 are involutions, listed by A014489.

Antti Karttunen, Table of n, a(n) for n = 0..5040

Main diagonal of A261096 and A261097.
Cf. A014489 (the positions of zeros).
Cf. A055089, A195663.
Cf. also A261219.
Related permutations: A060119, A060126.

a(n) = A261096(n,n) = A261097(n,n).
By conjugating a similar sequence:
a(n) = A060119(A261219(A060126(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))).

cp's OEIS Frontend

A060126 Positions of permutations of A055089 in the permutation sequence A060117.

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A060119 Positions of permutations of A060117 in reversed colexicographic ordering A055089.

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A261220 Ranks of involutions in permutation orderings A060117 and A060118.

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A261216 A(i,j) = rank (in A060117) of the composition of the i-th and the j-th permutation in table A060117, which lists all finite permutations.

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A261099 Main diagonal of A261096.

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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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