login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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, 1, 3, 2, 3, 12, 1, 11, 39, 11, 11, 116, 133, 12, 45, 449, 722, 169, 45, 996, 3857, 2832, 206 (list; graph; refs; listen; history; text; internal format)
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: A186102 A170848 A078017 * 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

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 26 09:38 EST 2020. Contains 338639 sequences. (Running on oeis4.)