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"Fourth down, Extream [sic] between the two farthest Bells from it" in bell-ringing is a sequence of permutations p_1=(1,2,3,4), p_2=(1,2,4,3), .. which runs through all permutations of {1,2,3,4} with period 24; sequence gives number in position 1 of n-th permutation.
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%I #18 Feb 27 2024 11:43:52

%S 1,1,1,4,4,1,1,1,3,3,3,4,4,3,3,3,2,2,2,4,4,2,2,2,1,1,1,4,4,1,1,1,3,3,

%T 3,4,4,3,3,3,2,2,2,4,4,2,2,2,1,1,1,4,4,1,1,1,3,3,3,4,4,3,3,3,2,2,2,4,

%U 4,2,2,2,1,1,1,4,4,1,1,1,3,3,3,4,4,3,3,3,2,2,2,4,4,2,2,2,1,1,1,4,4,1,1,1,3

%N "Fourth down, Extream [sic] between the two farthest Bells from it" in bell-ringing is a sequence of permutations p_1=(1,2,3,4), p_2=(1,2,4,3), .. which runs through all permutations of {1,2,3,4} with period 24; sequence gives number in position 1 of n-th permutation.

%C Start with (1,2,3,4), i.e. the first permutation of {1,2,3} followed by 4; then for each next permutation, transpose 4 one to the left; if at position 1, replace {1,2,3} recursively by the next permutation of these numbers. Thereafter, for each next permutation, transpose 4 to the right. And so on.

%H Richard Duckworth and Fabian Stedman, <a href="http://www.gutenberg.org/files/18567/18567-h/18567-h.htm">Tintinnalogia, or, the Art of Ringing</a>, (1671). Released by Project Gutenberg, 2006.

%H <a href="/index/Be#bell_ringing">Index entries for sequences related to bell ringing</a>

%H <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,1,0,0,-1,2,-2,1,0,0,-1,2,-2,1,0,0,-1,2,-2,1).

%F Period 24.

%e The full list of the 24 permutations is as follows (the present sequence gives the first column):

%e 1 2 3 4

%e 1 2 4 3

%e 1 4 2 3

%e 4 1 2 3

%e 4 1 3 2

%e 1 4 3 2

%e 1 3 4 2

%e 1 3 2 4

%e 3 1 2 4

%e 3 1 4 2

%e 3 4 1 2

%e 4 3 1 2

%e 4 3 2 1

%e 3 4 2 1

%e 3 2 4 1

%e 3 2 1 4

%e 2 3 1 4

%e 2 3 4 1

%e 2 4 3 1

%e 4 2 3 1

%e 4 2 1 3

%e 2 4 1 3

%e 2 1 4 3

%e 2 1 3 4

%p ring:= proc(k::nonnegint) local p,i,left,l,nf,ini; if k<=1 then proc() [1$k] end else ini := proc() p:= ring(k-1); i:= k; left:= true; l:= p(); nf:= k! end; ini(); proc() local ll; ll:= [seq(l[t], t=1..(i-1)), k, seq(l[t], t=i..(k-1))]; if left then if i>1 then i:= i-1 else left:= false; l:=p() fi else if i<k then i:= i+1 else left:= true; l:=p() fi fi; nf:= nf-1; if nf = 0 then ini() fi; ll end fi end: bell := proc(k) option remember; local p; p:= ring(k); [seq(p(), i=1..k!)] end: a := n-> bell(4)[modp(n-1,24)+1][1]: seq(a(n), n=1..121);

%t LinearRecurrence[{2, -2, 1, 0, 0, -1, 2, -2, 1, 0, 0, -1, 2, -2, 1, 0, 0, -1, 2, -2, 1} {1, 1, 1, 4, 4, 1, 1, 1, 3, 3, 3, 4, 4, 3, 3, 3, 2, 2, 2, 4, 4}, 105] (* _Jean-François Alcover_, Mar 14 2021 *)

%Y Cf. A143484-A143490, A090281.

%K nonn,easy

%O 1,4

%A _Alois P. Heinz_, Aug 19 2008