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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A060495 Each permutation in the list A060117 converted to Site Swap notation, with "zero throws" (fixed elements) replaced with n, the length of siteswap.

Original entry on oeis.org

1, 11, 312, 111, 231, 222, 4413, 1313, 4112, 1111, 2411, 2312, 4242, 1241, 4233, 1223, 2222, 2231, 3441, 3342, 3131, 3122, 3423, 3333, 55514, 14514, 51414, 11314, 25314, 24414, 55113, 14113, 51112, 11111, 25111, 24112, 52512, 12511, 52413
Offset: 0

Views

Author

Antti Karttunen, Mar 21 2001

Keywords

Comments

This sequence is not well-defined for n >= 3628800 because the Site Swap notation can contain values exceeding 9, for example, the Site Swap notation for a(3628800) is [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 10]. - Sean A. Irvine, Nov 25 2022

Links

Crossrefs

Cf. factorial base representation A007623 and A060496, A006694.
See also A060498, A060499, A061417. Average of digits gives number of balls: A060501.

Programs

  • Maple
    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;
SiteSwap1ToDec := proc(s) local i,z,n; n := nops(s); z := 0; for i from 1 to n do z := 10*z; if(0 = s[i]) then z := z+n; else z := z+s[i]; fi; od; RETURN(z); end;

Formula

a(n) = SiteSwap1ToDec(Perm2SiteSwap1(PermUnrank3R(n))).

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

Views

Author

Antti Karttunen, Mar 05 2001

Keywords

Links

Crossrefs

Cf. A055090, A056170, A060118, A060120, A060129, A060130, A275725, A278225, A276004.
Cf. A276005 (positions where coincides with A060502).

Programs

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

Formula

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)

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

Views

Author

Antti Karttunen, Aug 26 2015

Keywords

Comments

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)

Links

Crossrefs

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

Views

Author

Antti Karttunen, Aug 26 2015

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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.

Formula

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))).

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).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 11 2016

Keywords

Links

Crossrefs

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

Formula

a(n) = A056169(A275725(n)).
a(n) = 1 + A084558(n) - A060129(n).

A060132 Positions of the permutations which have the same rank in A055089 and A060117, i.e., the fixed points of permutations A060119 and A060126.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 16, 17, 24, 25, 26, 27, 30, 31, 32, 33, 40, 41, 60, 61, 62, 63, 120, 121, 122, 123, 126, 127, 128, 129, 136, 137, 144, 145, 146, 147, 150, 151, 152, 153, 160, 161, 180, 181, 182, 183, 288, 289, 290, 291, 294, 295, 296, 297, 304, 305, 316
Offset: 0

Views

Author

Antti Karttunen, Mar 02 2001

Keywords

Crossrefs

Cf. A060133. Includes A059590 as a subset and A064637 gives the terms that are not found therein.

Programs

  • Maple
    sub1 := n -> (n - 1); map(sub1,positions(0,[seq(PermRank3R(PermRevLexUnrank(n))-n,n=0..1024)])); or map(sub1,positions(0,[seq(PermRevLexRank(PermUnrank3R(n))-n,n=0..1024)]));

A065184 Permutation of nonnegative integers produced when the finite permutations listed by A060117 are subjected to the left-right maxima variant of Foata's transformation. Inverse of A065183.

Original entry on oeis.org

0, 1, 2, 5, 3, 4, 6, 7, 14, 23, 15, 22, 8, 21, 12, 16, 19, 11, 17, 10, 9, 13, 20, 18, 24, 25, 26, 29, 27, 28, 54, 55, 86, 119, 87, 118, 56, 117, 84, 88, 115, 59, 89, 58, 57, 85, 116, 114, 30, 31, 80, 107, 81, 106, 48, 49, 60, 67, 61, 66, 74, 92, 38, 113, 47, 101, 112, 100
Offset: 0

Views

Author

Antti Karttunen, Oct 19 2001

Keywords

Links

Crossrefs

A065161-A065163 give cycle counts and max lengths. Cf. also A065181, A065182 for Maple procedure Foata and A060117 for PermUnrank3R and A060125 for PermRank3R.

Programs

  • Maple
    [seq(PermRank3R(Foata(PermUnrank3R(j))),j=0..119)];

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

Views

Author

Antti Karttunen, Nov 16 2016

Keywords

Comments

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.

Links

Crossrefs

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.

Programs

Formula

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

A060496 Each permutation in the list A060117 converted to Site Swap notation, with digits reversed. "Zero throws" (fixed elements) indicated with 0's.

Original entry on oeis.org

0, 11, 210, 111, 102, 222, 3100, 3131, 2110, 1111, 1102, 2132, 2020, 1021, 3320, 3221, 2222, 1322, 1003, 2033, 1313, 2213, 3203, 3333, 41000, 41041, 41410, 41311, 41302, 41442, 31100, 31141, 21110, 11111, 11102, 21142, 21020, 11021, 31420
Offset: 0

Views

Author

Antti Karttunen, Mar 21 2001

Keywords

Comments

This sequence is not well-defined for n >= 3628800 because the Site Swap notation can contain values exceeding 9, for example, the Site Swap notation for a(3628800) is [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 10]. - Sean A. Irvine, Nov 25 2022

Links

Crossrefs

Cf. factorial base representation A007623 and A060495, A006694.
In A060498 the digits are also "inverted", giving valid siteswap juggling patterns.

Programs

  • Maple
    SiteSwap2ToDec := proc(s) local i,z; z := 0; for i from nops(s) by -1 to 1 do z := 10*z + s[i]; od; RETURN(z); end;

Formula

a(n) = SiteSwap2ToDec(Perm2SiteSwap1(PermUnrank3R(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

Views

Author

Antti Karttunen, Aug 11 2016

Keywords

Examples

			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.

Links

Crossrefs

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).

Programs

Formula

a(n) = A007814(A275725(n)).
Other identities:
For n >= 1, a(A033312(n)) = n.
For n >= 2, a(A000142(n)) = 1.
Previous Showing 11-20 of 54 results. Next

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