A060120 -id:A060120 - cp's OEIS frontend

A060127 Positions of permutations of A055089 in the permutation sequence A060118. Inverse permutation to A060120.

0, 1, 2, 5, 3, 4, 6, 7, 14, 23, 15, 22, 8, 11, 12, 19, 16, 21, 9, 10, 13, 18, 17, 20, 24, 25, 26, 29, 27, 28, 54, 55, 86, 119, 87, 118, 56, 59, 84, 115, 88, 117, 57, 58, 85, 114, 89, 116, 30, 31, 38, 47, 39, 46, 48, 49, 74, 101, 75, 100, 60, 67, 80, 107, 93, 112, 61, 66

Offset: 0

Antti Karttunen, Mar 02 2001

nonn

Index entries for sequences that are permutations of the natural numbers

Cf. A060125, A060126, A060133.

a(n) = PermRank3L(PermRevLexUnrank(n))

A060133 Positions of the permutations which have the same rank in A055089 and A060118, i.e., the fixed points of permutations A060120 and A060127.

Original entry on oeis.org

0, 1, 2, 6, 7, 16, 24, 25, 26, 60, 120, 121, 122, 126, 127, 136, 288, 289, 316, 450, 720, 721, 722, 726, 727, 736, 744, 745, 746, 780, 1680, 1681, 1682, 1812, 2592, 5040, 5041, 5042, 5046, 5047, 5056, 5064, 5065, 5066, 5100, 5160, 5161, 5162, 5166, 5167

Offset: 0

Antti Karttunen, Mar 02 2001

nonn

Cf. A060132.

map(sub1,positions(0,[seq(PermRevLexRank(PermUnrank3L(n))-n,n=0..6666)])); or map(sub1,positions(0,[seq(PermRank3L(PermRevLexUnrank(n))-n,n=0..6666)]));

A060118 A list of all finite permutations in "PermUnrank3L" ordering. (Inverses of the permutations of A060117.)

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 02 2001

In contrast to PermUnrank3R (A060117), PermUnrank3L applies each successive transposition from the left, not from the right, thus producing the inverse (permutation) of what PermUnrank3R would produce.

			In this table each row consists of A001563[n] permutations of (n+1) terms;
Append to each an infinite number of fixed terms and we get a list of rearrangements of natural numbers, but with only a finite number of terms permuted:
1/2,3,4,5,6,7,8,9,...
2,1/3,4,5,6,7,8,9,...
1,3,2/4,5,6,7,8,9,...
2,3,1/4,5,6,7,8,9,...
3,2,1/4,5,6,7,8,9,...
3,1,2/4,5,6,7,8,9,...
1,2,4,3/5,6,7,8,9,...
2,1,4,3/5,6,7,8,9,...

A060120 = Positions of these permutations in the "canonical list" A055089. Cf. also A060117.

with(group); permul := (a,b) -> mulperms(b,a); PermUnrank3L := proc(r) local n; n := nops(factorial_base(r)); convert(PermUnrank3Laux(n+1,r,[]),'permlist',1+(((r+2) mod (r+1))*n)); end; PermUnrank3Laux := proc(n,r,p) local s; if(0 = r) then RETURN(p); else s := floor(r/((n-1)!)); RETURN(PermUnrank3Laux(n-1, r-(s*((n-1)!)), permul([[n,n-s]],p))); fi; end;

[seq(op(PermUnrank3L(j)), j=0..)]; (Maple code given below)

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

A060131 a(n) = A072411(A275725(n)); order of each permutation listed in tables A060117 and A060118, i.e., the least common multiple of the cycle sizes.

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, 6, 2, 6, 3, 6, 4, 5, 4, 5, 3, 4, 4, 5, 6, 5, 3, 4, 6, 5, 4, 5, 2, 2, 3, 4, 3, 4, 3, 6, 4, 5, 4, 5, 2, 6, 4, 5, 6, 5, 2, 6, 6, 5, 4, 5, 2, 3, 3, 4, 2, 4, 2, 6, 4, 5, 6, 5, 3, 4, 4, 5, 6, 5, 2, 4, 6, 5, 6, 5, 2, 3, 2, 4, 3, 4, 2, 6, 6, 5, 4, 5, 2, 4, 6, 5, 6, 5, 3, 4, 6, 5, 4, 5, 2

Offset: 0

Antti Karttunen, Mar 05 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to lcm's

Cf. A055092, A060117, A060118, A060120, A072411, A275725.
Cf. A261220 (gives the positions of 1 and 2's).
Cf. A275813 (indices of odd terms), A275814 (indices of even terms).

(define (A060131 n) (A072411 (A275725 n)))

From Antti Karttunen, Aug 09 2016: (Start)
a(n) = A072411(A275725(n)).
a(n) = A055092(A060120(n)).
(End)

A060129 Number of moved (non-fixed) elements in the permutation with rank number n in lists A060117 (or in A060118), i.e., the sum of the lengths of all cycles larger than one in that permutation.

Original entry on oeis.org

0, 2, 2, 3, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 3, 4, 4, 4, 2, 3, 4, 4, 3, 4, 2, 4, 4, 5, 4, 5, 3, 5, 4, 5, 4, 5, 3, 4, 4, 5, 5, 5, 3, 4, 5, 5, 4, 5, 2, 4, 3, 4, 3, 4, 3, 5, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 4, 5, 5, 5, 4, 5, 2, 3, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 3, 4, 4, 5, 5, 5, 4, 4, 5, 5, 5, 5, 2, 3, 4, 4, 3, 4, 4, 5, 5, 5, 4, 5, 4, 4, 5, 5, 5, 5, 3, 4, 5, 5, 4, 5, 2

Offset: 0

Antti Karttunen, Mar 05 2001

nonn

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

Cf. also A000142, A055093, A060117, A060118, A060120, A060128, A060130, A084558, A275851.

a(n) = A060128(n) + A060130(n).
From Antti Karttunen, Aug 11 2016: (Start)
a(n) = A275812(A275725(n)).
a(n) = 1 + A084558(n) - A275851(n).
Other identities. For all n >= 0:
a(n) = A055093(A060120(n)).
a(A000142(n)) = 2.
(End)

A060128 a(n) is the number of disjoint cycles (excluding 1-cycles, i.e., fixed elements) in the n-th permutation of A060117 and A060118.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 05 2001

nonn

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

Cf. A055090, A056170, A060118, A060120, A060129, A060130, A275725, A278225, A276004.

Cf. A276005 (positions where coincides with A060502).

A060128(n) = nops(convert(PermUnrank3L(n), 'disjcyc')); # Code for function PermUnrank3L given in A060118.

a(n) = A060129(n) - A060130(n).
From Antti Karttunen, Aug 07 2017: (Start)
a(n) = A056170(A275725(n)).
a(n) = A055090(A060120(n)).
a(n) = A060502(n) - A276004(n).
(End)

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.

Original entry on oeis.org

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))).

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))).

cp's OEIS Frontend

A060127 Positions of permutations of A055089 in the permutation sequence A060118. Inverse permutation to A060120.

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A060133 Positions of the permutations which have the same rank in A055089 and A060118, i.e., the fixed points of permutations A060120 and A060127.

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A060118 A list of all finite permutations in "PermUnrank3L" ordering. (Inverses of the permutations of A060117.)

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

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A060131 a(n) = A072411(A275725(n)); order of each permutation listed in tables A060117 and A060118, i.e., the least common multiple of the cycle sizes.

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A060129 Number of moved (non-fixed) elements in the permutation with rank number n in lists A060117 (or in A060118), i.e., the sum of the lengths of all cycles larger than one in that permutation.

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A060128 a(n) is the number of disjoint cycles (excluding 1-cycles, i.e., fixed elements) in the n-th permutation of A060117 and A060118.

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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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Formula

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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Formula

A261097 Transpose of square array A261096.