A261217 -id:A261217 - cp's OEIS frontend

A261096 A(i,j) = rank (in A055089) of the composition of the i-th and the j-th permutation in table A055089, which lists all finite permutations in reversed colexicographic ordering.

0, 1, 1, 2, 0, 2, 3, 4, 3, 3, 4, 5, 0, 2, 4, 5, 2, 1, 5, 5, 5, 6, 3, 5, 4, 1, 4, 6, 7, 7, 4, 0, 0, 3, 7, 7, 8, 6, 12, 1, 3, 2, 8, 6, 8, 9, 10, 13, 13, 2, 1, 9, 10, 9, 9, 10, 11, 14, 12, 18, 0, 10, 11, 6, 8, 10, 11, 8, 15, 16, 19, 19, 11, 8, 7, 11, 11, 11, 12, 9, 16, 17, 20, 18, 0, 9, 11, 10, 7, 10, 12, 13, 18, 17, 14, 21, 22, 1, 1, 10, 6, 6, 9, 13, 13, 14, 19, 6, 15, 22, 23, 2, 0, 14, 7, 9, 8, 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 (in ordering used by table A055089) 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 A055089 (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...
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,  4,  5,  2,  3,  7,  6, 10, 11,  8,  9, 18, ...
   2,  3,  0,  1,  5,  4, 12, 13, 14, 15, 16, 17,  6, ...
   3,  2,  5,  4,  0,  1, 13, 12, 16, 17, 14, 15, 19, ...
   4,  5,  1,  0,  3,  2, 18, 19, 20, 21, 22, 23,  7, ...
   5,  4,  3,  2,  1,  0, 19, 18, 22, 23, 20, 21, 13, ...
   6,  7,  8,  9, 10, 11,  0,  1,  2,  3,  4,  5, 14, ...
   7,  6, 10, 11,  8,  9,  1,  0,  4,  5,  2,  3, 20, ...
   8,  9,  6,  7, 11, 10, 14, 15, 12, 13, 17, 16,  0, ...
   9,  8, 11, 10,  6,  7, 15, 14, 17, 16, 12, 13, 21, ...
  10, 11,  7,  6,  9,  8, 20, 21, 18, 19, 23, 22,  1, ...
  11, 10,  9,  8,  7,  6, 21, 20, 23, 22, 18, 19, 15, ...
  12, 13, 14, 15, 16, 17,  2,  3,  0,  1,  5,  4,  8, ...
  ...
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 A055089, 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 4th one in A055089, thus A(1,2) = 4.
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 A055089, 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: A261097.
Row 0 & Column 0: A001477 (identity permutation).
Row 1: A261098.
Column 1: A004442.
Main diagonal: A261099.
Cf. tables A055089, A195663.
Cf. also A261216, A261217 (similar arrays, but using different orderings of permutations).
Permutations used in conjugation-formulas: A056019, A060119, A060120, A060126, A060127.

By conjugating with related permutations and arrays:
A(i,j) = A056019(A261097(A056019(i),A056019(j))).
A(i,j) = A060119(A261216(A060126(i),A060126(j))).
A(i,j) = A060120(A261217(A060127(i),A060127(j))).

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))).

A261219 Main diagonal of A261216: a(n) = A261216(n,n).

Original entry on oeis.org

0, 0, 0, 5, 0, 3, 0, 0, 14, 16, 22, 20, 0, 19, 8, 20, 0, 7, 0, 13, 0, 7, 10, 16, 0, 0, 0, 5, 0, 3, 54, 54, 60, 65, 66, 69, 84, 90, 78, 95, 84, 81, 114, 108, 114, 107, 102, 111, 0, 0, 74, 76, 100, 98, 30, 30, 78, 83, 102, 105, 0, 19, 26, 45, 100, 119, 0, 13, 74, 87, 28, 41, 0, 97, 50, 98, 0, 49, 0, 97, 26, 117, 22, 47, 36, 108, 60, 113, 36, 63, 0, 25, 50, 33, 10, 59, 0, 73, 0, 49, 52

Offset: 0

Antti Karttunen, Aug 26 2015

nonn

Equally: main diagonal of A261217.

For permutation p, which has rank n in permutation list A060117, 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. Equally, if permutation p has rank n in the order used in list A060118, a(n) gives the rank of the p*p in that same list. Thus zeros (which mark the identity permutation, with rank 0 in both orders) occur at positions where the permutations of A060117 (equally: of A060118) are involutions, listed by A261220.

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

Main diagonal of A261216 and A261217.
Cf. A261220 (the positions of zeros).
Cf. A060117, A060118.
Cf. also A261099, A089841.
Related permutations: A060119, A060126.

a(n) = A261216(n,n) = A261217(n,n).
By conjugating a similar sequence:
a(n) = A060126(A261099(A060119(n))).

A261097 Transpose of square array A261096.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 26 2015

Each row and column is a permutation of A001477. See the comments at A261096.

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

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

Transpose: A261096.
Row 0 & Column 0: A001477 (identity permutation).
Row 1: A004442.
Column 1: A261098.
Main diagonal: A261099.
Cf. also A055089, A195663.
Cf. also A261216, A261217 (similar arrays, but using different orderings of permutations).
Permutations used in conjugation-formulas: A056019, A060119, A060120, A060126, A060127.

By conjugating with related permutations and arrays:
A(i,j) = A056019(A261096(A056019(i),A056019(j))).
A(i,j) = A060119(A261217(A060126(i),A060126(j))).
A(i,j) = A060120(A261216(A060127(i),A060127(j))).

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

A261096 A(i,j) = rank (in A055089) of the composition of the i-th and the j-th permutation in table A055089, which lists all finite permutations in reversed colexicographic ordering.

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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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A261219 Main diagonal of A261216: a(n) = A261216(n,n).

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A261097 Transpose of square array A261096.

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A261218 Row 1 of A261216.

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