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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A060134 Positions of permutations produced by the transposition sequence A060135 in A055089.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Mar 02 2001

Keywords

Crossrefs

Cf. A057113.

Programs

  • Maple
    apply_transpositions(A060135); apply_transpositions := proc(a) local k,b,t,p; p := []; k := nops(factorial_base(nops(a)))+1; b := [PermRevLexRank(convert(p,'permlist',k))]; for t in a do p := permul(p,[[t,t+1]]); b := [op(b),PermRevLexRank(convert(p,'permlist',k))]; od; RETURN(b); end;

A057112 Sequence of 719 adjacent transpositions (a[n] a[n]+1), which, when starting from the identity permutation and applied successively, produce a Hamiltonian circuit/path through all 720 permutations of S_6, in such a way that S_{n-1} is always traversed before the rest of S_n.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 3, 2, 3, 4, 1, 2, 1, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 3, 2, 3, 4, 1, 2, 1, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 3, 2, 3, 4, 1, 2, 1, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 3, 2, 3, 4, 1, 2, 1, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 3, 2, 3
Offset: 1

Views

Author

Antti Karttunen, Aug 09 2000

Keywords

Comments

If the 120 permutations of S_5 are connected by adjacent transpositions, the graph produced is isomorphic to the prismatodecachoron (a 4-dimensional polytope) graph (see the Olshevsky link) and this sequence gives directions for a Hamiltonian circuit through its vertices. The first 24 terms give a Hamiltonian path through truncated octahedron's graph (the last path shown in the Karttunen link).
Comment from N. J. A. Sloane: This is the subject of "bell-ringing" or "change-ringing", which has been studied for hundreds of years. See for example Amer. Math. Monthly, Vol. 94, Number 8, 1987, pp. 721-.

Examples

			Starting from the identity permutation and applying these transpositions (from right), we get:
[1,2,3,4,5,6,...] o (1 2) ->
[2,1,3,4,5,6,...] o (2 3) ->
[2,3,1,4,5,6,...] o (1 2) ->
[3,2,1,4,5,6,...] o (2 1) ->
[3,1,2,4,5,6,...] o (1 2) ->
[1,3,2,4,5,6,...] o (3 4) ->
[1,3,4,2,5,6,...] o (1 2) ->
[3,1,4,2,5,6,...] o (2 3) ->
[3,4,1,2,5,6,...] o (3 4) etc.

Links

Crossrefs

Cf. A057113, A055089 (for the Maple definitions of fac_base and cdr), A060135 (palindromic variant of the same idea).

Programs

  • Maple
    adj_tp_seq := proc(n) local fl,fd,v; fl := fac_base(n); fd := fl[1]; if((1 = fd) and (0 = convert(cdr(fl),`+`))) then RETURN(nops(fl)); fi; if(n < 6) then RETURN(2 - (`mod`(n,2))); fi; if((0 = convert(cdr(fl),`+`)) and (n < 24)) then RETURN((nops(fl)+1)-fd); fi; if(n < 18) then if(0 = (`mod`(n,2))) then RETURN(2); else RETURN(4-(`mod`(n,4))); fi; else if(n < 24) then RETURN(2+(`mod`(n,2))); else if(n < 120) then if(0 = convert(cdr(fl),`+`)) then RETURN(nops(fl)); else RETURN(adj_tp_seq(`mod`(n,24))); fi; else if(n < 720) then if(125 = n) then RETURN(5); fi; v := (`mod`(n,5)); if(0 = v) then v := (n-125)/5; RETURN(adj_tp_seq(v)+(`mod`(v+1,2))); else if(5 > (`mod`(n,10))) then RETURN(5-v); else RETURN(v); fi; fi; else if(0 = convert(cdr(fl),`+`)) then RETURN(nops(fl)); fi; RETURN(adj_tp_seq(`mod`(n,720))); fi; fi; fi; fi; end;

Formula

tp_seq := [seq(adj_tp_seq(n), n=1..719)];
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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