A169816 Triangle read by rows: T(n,k) is the number of down-up permutations of {1,2,...,n} having genus k.

1, 1, 1, 1, 3, 2, 3, 12, 1, 11, 39, 11, 11, 116, 133, 12, 45, 449, 722, 169, 45, 996, 3857, 2832, 206

Offset

1,5

Comments

The genus g(p) of a permutation p of {1,2,...,n} is defined by g(p)=(1/2)[n+1-z(p)-z(cp')], where p' is the inverse permutation of p, c = 234...n1 = (1,2,...,n), and z(q) denotes the number of cycles of the permutation q.

The sum of the entries in row n is A000111(n) (Euler or up-down numbers).

Apparently, row n contains ceiling(n/2) entries.

T(2n,0) = T(2n+1,0) = A001003(n) (the little Schroeder numbers).

The Maple program yields the entries of row n (specified at the start of the program).

Links

Table of n, a(n) for n=1..25.

S. Dulucq and R. Simion, Combinatorial statistics on alternating permutations, J. Algebraic Combinatorics, 8, 1998, 169-191.

Example

T(3,1)=1 because 312 is the only down-up permutation of {1,2,3} with genus 1 (we have p=312=(132), cp'=231*231=312=(132) and so g(p) = (1/2)(3+1-1-1) = 1).
Triangle starts:
   1;
   1;
   1,   1;
   3,   2;
   3,  12,   1;
  11,  39,  11;
  11, 116, 133,  12;

Maple

n := 6: with(combinat): descents := proc (p) local A, i: A := {}: for i to nops(p)-1 do if p[i+1] < p[i] then A := `union`(A, {i}) else end if end do: A end proc:
DU := proc (n) local du, P, j: du := {}: P := permute(n): for j to factorial(n) do if descents(P[j]) = {seq(2*k-1, k = 1 .. floor((1/2)*n))} then du := `union`(du, {P[j]}) else end if end do: du end proc:
inv := proc (p) local j, q: for j to nops(p) do q[p[j]] := j end do: [seq(q[i], i = 1 .. nops(p))] end proc:
nrcyc := proc (p) local nrfp, pc: nrfp := proc (p) local ct, j: ct := 0: for j to nops(p) do if p[j] = j then ct := ct+1 else end if end do; ct end proc:
pc := convert(p, disjcyc): nops(pc)+nrfp(p) end proc:
b := proc (p) local c; c := [seq(i+1, i = 1 .. nops(p)-1), 1]: [seq(c[p[j]], j = 1 .. nops(p))] end proc:
gen := proc (p) options operator, arrow: (1/2)*nops(p)+1/2-(1/2)*nrcyc(p)-(1/ 2)*nrcyc(b(inv(p))) end proc: f[n] := sort(add(t^gen(DU(n)[j]), j = 1 .. nops(DU(n)))): seq(coeff(f[n], t, j), j = 0 .. ceil((1/2)*n)-1); # yields the entries of row n (specified at the start of the program)

Crossrefs

Cf. A000111, A001003.

Sequence in context: A170848 A078017 A343170 * A291739 A057053 A081850

Adjacent sequences: A169813 A169814 A169815 * A169817 A169818 A169819

Keyword

more,nonn,tabf

Author

Emeric Deutsch, May 28 2010

Extensions

Edited by R. J. Mathar, Jun 08 2010

Status

approved