cp's OEIS Frontend

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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A055092 Order of each permutation given in reversed colexicographic ordering A055089, i.e., the least common multiple of their cycle lengths.

Original entry on oeis.org

1, 2, 2, 3, 3, 2, 2, 2, 3, 4, 4, 3, 3, 4, 2, 3, 2, 4, 4, 3, 3, 2, 4, 2, 2, 2, 2, 6, 6, 2, 3, 6, 4, 5, 5, 4, 4, 5, 3, 4, 6, 5, 5, 4, 4, 3, 5, 6, 3, 6, 4, 5, 5, 4, 2, 2, 3, 4, 4, 3, 2, 6, 4, 5, 5, 6, 6, 2, 5, 4, 6, 5, 4, 5, 3, 4, 6, 5, 3, 4, 2, 3, 2, 4, 4, 5, 2, 6, 6, 5, 5, 6, 6, 5, 2, 4, 5, 4, 4, 3, 5, 6, 4, 3, 3
Offset: 0

Views

Author

Antti Karttunen, Apr 04 2000

Keywords

Links

Crossrefs

Cf. A055089, A060131, A072411, A290095.

Programs

  • Maple
    A055092(n) = count_permorder(convert(PermRevLexUnrank(j), 'disjcyc')).
count_permorder := proc(l) local c,t; t := 1; for c in l do t := ilcm(t,nops(c)); od; RETURN(t); end;
# Procedure PermRevLexUnrank given in A055089.

Formula

a(n) = A072411(A290095(n)) = A060131(A060126(n)). - Antti Karttunen, Dec 30 2017

Extensions

Entry revised by Antti Karttunen, Dec 30 2017

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)]));

A055090 Number of cycles (excluding fixed points) of the n-th finite permutation in reversed colexicographic ordering (A055089).

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, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2
Offset: 0

Views

Author

Antti Karttunen, Apr 18 2000

Keywords

Comments

Among the first n! entries k appears A136394(n,k) times. - Tilman Piesk, Apr 06 2012

Links

Crossrefs

Cf. A195663, A195664, A055089 (ordered finite permutations).
Cf. A198380 (cycle type of the n-th finite permutation).
Cf. A136394, A055089, A055090, A055093, A055091, A060128, A290095.

Programs

  • Maple
    with(group); seq(nops(convert(PermRevLexUnrank(j),'disjcyc')),j=0..)];
# Procedure PermRevLexUnrank given in A055089.

Formula

a(n) = A055093(n) - A055091(n).
a(n) = A056170(A290095(n)) = A060128(A060126(n)). - Antti Karttunen, Dec 30 2017

Extensions

Name changed by Tilman Piesk, Apr 06 2012

A057113 Positions of permutations produced by the transposition sequence A057112 in A055089.

Original entry on oeis.org

0, 1, 4, 5, 3, 2, 12, 13, 16, 22, 19, 18, 20, 10, 7, 6, 8, 14, 15, 9, 11, 21, 23, 17, 77, 76, 73, 72, 74, 75, 85, 84, 86, 80, 78, 79, 82, 92, 90, 91, 94, 88, 89, 95, 93, 83, 81, 87, 63, 62, 60, 61, 64, 65, 71, 70, 67, 53, 51, 50, 48, 54, 56, 57, 59, 69, 68, 58, 55, 49, 52, 66, 108, 109, 112, 113, 111, 110, 104, 105, 107, 117, 119, 118, 115, 101, 99, 98, 96, 102
Offset: 0

Views

Author

Antti Karttunen, Aug 09 2000

Keywords

Links

Crossrefs

PermRevLexRank given in A056019.

Programs

  • Maple
    atp_perm_ranks := proc(upto_n) local t,a,p,i,k; p := convert([1],'disjcyc'); k := nops(factorial_base(upto_n))+1; a := []; for i from 1 to upto_n do a := [op(a),PermRevLexRank(convert(p,'permlist',k))]; t := adj_tp_seq(i); p := mulperms([[t,t+1]],p); od; RETURN(a); end;

Formula

perm_ranks_seq := atp_perm_ranks(120);

A014489 Positions of involutions (permutations whose square is the identity) in reverse colexicographic order (A055089/A195663).

Original entry on oeis.org

0, 1, 2, 5, 6, 7, 14, 16, 21, 23, 24, 25, 26, 29, 54, 55, 60, 67, 80, 82, 86, 94, 105, 107, 111, 119, 120, 121, 122, 125, 126, 127, 134, 136, 141, 143, 264, 265, 266, 269, 288, 289, 314, 316, 339, 341, 390, 391, 396, 403, 414, 415, 444, 450, 469
Offset: 0

Views

Author

Wouter Meeussen

Keywords

Links

Crossrefs

Positions of zeros in A261099.
From a(1)=1 onward also positions of 2's in A055092.
Subsequences: A060112, A064640.
Cf. A055089, A195663.
Cf. also A261220.

Programs

  • Maple
    N:= 100: # to get a(0) to a(N)
M:= 0: A[0]:= 0: count:= 0:
for m from 2 while count < N do
  P:= remove(t -> t[1]=1, combinat:-permute(m));
  P:= map(t -> ListTools:-Reverse(subs([seq(i=m+1-i,i=1..m)],t)),P);
  R:= select(t -> max(map(nops,convert(P[t],disjcyc))) = 2, [$1..nops(P)]);
  for r in R do
     count:= count+1;
     A[count]:= r+M;
     if count = N then break fi;
  od:
  M:= M + nops(P);
od:
seq(A[i],i=0..count); # Robert Israel, Oct 28 2015

Extensions

Name changed by Antti Karttunen, Aug 30 2015

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

Views

Author

Antti Karttunen, Mar 02 2001

Keywords

Crossrefs

Cf. A060132.

Programs

  • Maple
    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)]));

A064638 Positions of non-crossing fixed-point-free involutions encoded by A014486 in A055089. Permutation of A064640.

Original entry on oeis.org

0, 1, 7, 23, 127, 415, 143, 659, 719, 5167, 16687, 5455, 26815, 28495, 5183, 16703, 5699, 36899, 38579, 5759, 36959, 40031, 40319, 368047, 1174447, 379567, 1901647, 1992367, 368335, 1174735, 389695, 2627455, 2718175, 391375, 2629135, 2799055
Offset: 0

Views

Author

Antti Karttunen, Oct 02 2001

Keywords

Crossrefs

Maple procedure binexp2pars given in A057501, permul in A060125.

Programs

  • Maple
    map(PermRevLexRank,map(NonCrossingTranspos, A014486));
NonCrossingTranspos := n -> convert(NonCrossingTransposAux(binexp2pars(n),1),'permlist',binwidth(n));
NonCrossingTransposAux := proc(s,ii) local e,p,i,j; i := ii; p := []; for e in s do p := permul(p,NonCrossingTransposAux(e,i+1)); j := i+CountParens(e)+1; p := permul(p,[[i,j]]); i := j+1; od; RETURN(p); end;
CountParens := proc(s) local e,k; if(0 = nops(s)) then RETURN(0); fi; e := 0; for k in s do e := e+2+CountParens(k); od; RETURN(e); end;

A065182 Permutation of nonnegative integers produced when the finite permutations listed by A055089 are subjected to Foata transform. Inverse of A065181.

Original entry on oeis.org

0, 1, 2, 4, 5, 3, 6, 7, 12, 18, 19, 13, 14, 16, 8, 22, 20, 10, 21, 23, 11, 17, 15, 9, 24, 25, 26, 28, 29, 27, 48, 49, 72, 96, 97, 73, 74, 76, 50, 100, 98, 52, 99, 101, 53, 77, 75, 51, 54, 55, 60, 66, 67, 61, 30, 31, 84, 108, 109, 85, 78, 91, 36, 115, 102, 42, 103, 114, 43
Offset: 0

Views

Author

Antti Karttunen, Oct 19 2001

Keywords

Comments

Here we use a variant of Foata's transformation, which forms a new permutation by "inserting parentheses" at each left-right maxima, to delimit cycles.

References

  • I. M. Gessel and R. P. Stanley, Algebraic Enumeration, chapter 21 in Handbook of Combinatorics, Vol. 2, edited by R.L.Graham et al., The MIT Press, Mass, 1995, page 1045.

Links

Crossrefs

A065161-A065163 give cycle counts and max lengths. Cf. also A065183, A065184 and A055089 and A056019 for the requisite Maple procedures.

Programs

  • Maple
    [seq(PermRevLexRank(Foata(PermRevLexUnrank(j))),j=0..119)];
with(group); Foata := proc(p) local c,c1,i,m; c := []; c1 := []; m := 0; for i from 1 to nops(p) do if(p[i] > m) then if(nops(c1) > 1) then c := [op(c),c1]; fi; m := p[i]; c1 := []; fi; c1 := [op(c1),p[i]]; od; if(nops(c1) > 1) then c := [op(c),c1]; fi; RETURN(convert(c,'permlist',nops(p))); end;

A290096 Filter-sequence related to cycle-structure of permutations listed in table A055089: Least number with the same prime signature as A290095.

Original entry on oeis.org

2, 4, 12, 8, 8, 12, 60, 36, 24, 16, 16, 24, 24, 16, 60, 24, 36, 16, 16, 24, 24, 60, 16, 36, 420, 180, 180, 72, 72, 180, 120, 72, 48, 32, 32, 48, 48, 32, 120, 48, 72, 32, 32, 48, 48, 120, 32, 72, 120, 72, 48, 32, 32, 48, 420, 180, 120, 48, 48, 120, 180, 72, 48, 32, 32, 72, 72, 180, 32, 48, 72, 32, 48, 32, 120, 48, 72, 32
Offset: 0

Views

Author

Antti Karttunen, Aug 17 2017

Keywords

Links

Crossrefs

Cf. A046523, A060126, A290095, A290097 (rgs-transform of this sequence).
Other filter-sequences related to factorial base and finite permutations: A278225, A278234, A278235, A278236.

Formula

a(n) = A046523(A290095(n)).
a(n) = A278225(A060126(n)).

A290097 Restricted growth sequence transform of A290096, related to cycle-structure of permutations listed in table A055089.

Original entry on oeis.org

1, 2, 3, 4, 4, 3, 5, 6, 7, 8, 8, 7, 7, 8, 5, 7, 6, 8, 8, 7, 7, 5, 8, 6, 9, 10, 10, 11, 11, 10, 12, 11, 13, 14, 14, 13, 13, 14, 12, 13, 11, 14, 14, 13, 13, 12, 14, 11, 12, 11, 13, 14, 14, 13, 9, 10, 12, 13, 13, 12, 10, 11, 13, 14, 14, 11, 11, 10, 14, 13, 11, 14, 13, 14, 12, 13, 11, 14, 12, 13, 9, 12, 10, 13, 13, 14, 10, 11, 11, 14, 14, 11, 11, 14, 10, 13, 14
Offset: 0

Views

Author

Antti Karttunen, Aug 17 2017

Keywords

Links

Crossrefs

Cf. A290095, A290096.
Cf. tables A055089, A195663.
Previous Showing 11-20 of 70 results. Next

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