A056019 -id:A056019 - cp's OEIS frontend

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

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

Antti Karttunen, Aug 09 2000

Sean A. Irvine, Table of n, a(n) for n = 0..119
Sean A. Irvine, Complete Maple program with all necessary functions

PermRevLexRank given in A056019.

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;

perm_ranks_seq := atp_perm_ranks(120);

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

Antti Karttunen, Oct 19 2001

nonn

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.

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.

Joe Buhler and R. L. Graham, Juggling Drops and Descents, Amer. Math. Monthly, 101, (no. 6) 1994, 507 - 519.
Index entries for sequences that are permutations of the natural numbers

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

[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;

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

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