A177267 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having genus k (see first comment for definition of genus).

1, 2, 0, 5, 1, 0, 14, 10, 0, 0, 42, 70, 8, 0, 0, 132, 420, 168, 0, 0, 0, 429, 2310, 2121, 180, 0, 0, 0, 1430, 12012, 20790, 6088, 0, 0, 0, 0, 4862, 60060, 174174, 115720, 8064, 0, 0, 0, 0, 16796, 291720, 1309308, 1624480, 386496, 0, 0, 0, 0, 0, 58786, 1385670, 9087078, 18748730, 10031736, 604800, 0, 0, 0, 0, 0, 208012, 6466460, 59306676, 188208020, 186698512, 38113920, 0

Offset: 1

Author

Emeric Deutsch, May 27 2010

Keywords

• nonn
• tabl

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) is the number of cycles of the permutation q.

The sum of the entries in row n is n!.

The number of nonzero entries in row n is floor((n+1)/2).

T(n,0) = A000108(n) (the Catalan numbers).

Apparently T(n,1) = A002802(n-3).

Last nonzero terms in rows with odd n appear to be A060593. [Joerg Arndt, Nov 01 2012.]

Examples

			T(3,1)=1 because 312 is the only 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,
[ 2]  2, 0,
[ 3]  5, 1, 0,
[ 4]  14, 10, 0, 0,
[ 5]  42, 70, 8, 0, 0,
[ 6]  132, 420, 168, 0, 0, 0,
[ 7]  429, 2310, 2121, 180, 0, 0, 0,
[ 8]  1430, 12012, 20790, 6088, 0, 0, 0, 0,
[ 9]  4862, 60060, 174174, 115720, 8064, 0, 0, 0, 0,
[10]  16796, 291720, 1309308, 1624480, 386496, 0, 0, 0, 0, 0,
[11]  58786, 1385670, 9087078, 18748730, 10031736, 604800, 0, 0, ...,
[12]  208012, 6466460, 59306676, 188208020, 186698512, 38113920, 0, ...,
[13]  742900, 29745716, 368588220, 1700309468, 2788065280, 1271140416, 68428800, 0, ...,
...
		

References

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

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Cf. A178514 (genus of derangements), A178515 (genus of involutions), A178516 (genus of up-down permutations), A178517 (genus of non-derangement permutations), A178518 (permutations of [n] having genus 0 and p(1)=k), A185209 (genus of connected permutations), A218538 (genus of permutations avoiding [x,x+1]).

Row sums give A000142.

T(2n+1,n) gives A060593.

Cf. A000108, A002802.

Programs

Maple

n := 8: with(combinat): P := permute(n): 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(P[j]), j = 1 .. factorial(n))): seq(coeff(f[n], t, j), j = 0 .. ceil((1/2)*n)-1); # yields the entries in the specified row n
# second Maple program:
b:= proc(n) option remember; `if`(n<2, 1, ((4*n-2)*
      b(n-1)+(n-2)*(n-1)^2*expand(x*b(n-2)))/(n+1))
    end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n, k), k=0..n-1), n=1..12);  # Alois P. Heinz, Feb 16 2024

Mathematica

T[ n_, k_] := If[ n < 1 || k >= n, 0, Module[{pn = Table[i, {i, n}]}, Do[ pn[[i]] = ((4 i - 2) pn[[i - 1]] + x (i - 2) (i - 1)^2 pn[[i - 2]])/(i + 1) // Expand, {i, 3, n}]; Coefficient[pn[[n]], x, k]]]; (* Michael Somos, Sep 02 2017 *)

Formula

Let p(n, x) := g.f. of row n. Then (n+1) * p(n, x) = (4*n-2) * p(n-1, x) + x * (n-2) * (n-1)^2 * p(n-2, x). - Michael Somos, Sep 02 2017

Extensions

Terms for rows 12 and 13 from Joerg Arndt, Jan 24 2011.