login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 way that S_{n-1} is always traversed before the rest of S_n. 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, 4, 1, 2, 1, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 3, 2, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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-.

LINKS

Table of n, a(n) for n=1..119.

A. Karttunen, Truncated octahedron

G. Olshevsky, Great prismatodecachoron

Index entries for sequences related to bell ringing

FORMULA

tp_seq := [seq(adj_tp_seq(n), n=1..719)];

EXAMPLE

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.

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;

CROSSREFS

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

Sequence in context: A263274 A292997 A060135 * A071956 A077767 A263259

Adjacent sequences:  A057109 A057110 A057111 * A057113 A057114 A057115

KEYWORD

nonn,fini

AUTHOR

Antti Karttunen, Aug 09 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 13 09:55 EST 2017. Contains 295957 sequences.