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

0, 1, 2, 5, 3, 4, 6, 7, 14, 23, 17, 20, 8, 11, 12, 22, 13, 21, 9, 10, 16, 18, 15, 19, 24, 25, 26, 29, 27, 28, 54, 55, 86, 119, 95, 110, 62, 71, 78, 116, 79, 113, 65, 68, 92, 102, 89, 103, 30, 31, 38, 47, 41, 44, 48, 49, 84, 118, 94, 108, 50, 53, 80, 117, 83, 109, 51, 52

Offset: 0

Author

Antti Karttunen, Oct 19 2001

Keywords

• nonn

Comments

Here we use the inverse of the left-right maxima variant of Foata's transformation, which works by rotating each cycle largest element first and then sorts the cycles to ascending order, according to that first (and largest) element of each.

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

• 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

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(FoataInv(PermRevLexUnrank(j))),j=0..119)];
with(group); FoataInv := p -> map(op, sort([op(map(RotCycleLargestFirst,convert(p,`disjcyc`))),op(FixedCycles(p))], sortbyfirst));
sortbyfirst := (a,b) -> `if`((a[1] < b[1]),true,false);
FindLargest := proc(a) local i,m; m := 0; for i from 1 to nops(a) do if(0 = m) then m := i; else if(a[i] > a[m]) then m := i; fi; fi; od; RETURN(m); end;
RotCycleLargestFirst := proc(c) local x; x := FindLargest(c); if(x <= 1) then RETURN(c); else RETURN([op(c[x..nops(c)]),op(c[1..(x-1)])]); fi; end;
FixedCycles := proc(p) local a,i; a := []; for i from 1 to nops(p) do if(p[i] = i) then a := [op(a),[i]]; fi; od; RETURN(a); end;