A178518 Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} having genus 0 and such that p(1)=k (see first comment for definition of genus).

1, 1, 1, 2, 2, 1, 5, 5, 2, 2, 14, 14, 5, 4, 5, 42, 42, 14, 10, 10, 14, 132, 132, 42, 28, 25, 28, 42, 429, 429, 132, 84, 70, 70, 84, 132, 1430, 1430, 429, 264, 210, 196, 210, 264, 429, 4862, 4862, 1430, 858, 660, 588, 588, 660, 858, 1430, 16796, 16796, 4862, 2860, 2145, 1848, 1764, 1848, 2145, 2860, 4862

Offset

1,4

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.

Row sums are A000108 (Catalan numbers).

T(n,1) = A000108(n-1) (n >= 1).

T(n,2) = A000108(n-1) (n >= 2).

T(n,3) = A000108(n-2) (n >= 3).

T(n,n) = A000108(n-2) (n >= 2).

A permutation p of {1,2,...,n} has genus 0 if and only if the cycle decomposition of p gives a noncrossing partition of {1,2,...,n} and each cycle of p is increasing (see Lemma 2.1 of the Dulucq-Simion reference).

References

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

Links

Table of n, a(n) for n=1..66.

Formula

T(n,1)=c(n-1); T(n,k) = c(n-k+1)*c(k-2) if 2 <= k <= n, where c(j) = binomial(2j,j)/(j+1) = A000108(j) are the Catalan numbers.

G.f. = G(t,z) = t*z*C(z)+t^2*z*(C(z)-1)*C(tz), where C(z) = (1-sqrt(1-4*z))/(2z) is the Catalan function.

Example

T(4,3)=2 because we have 3214=(13)(2)(4) and 3241=(134)(2).
Triangle starts:
   1;
   1,  1;
   2,  2,  1;
   5,  5,  2,  2;
  14, 14,  5,  4,  5;

Maple

c := proc (n) options operator, arrow: binomial(2*n, n)/(n+1) end proc: a := proc (n, k) if k = 1 then c(n-1) elif k <= n then c(n-k+1)*c(k-2) else 0 end if end proc: for n to 11 do seq(a(n, k), k = 1 .. n) end do; # yields sequence in triangular form

Mathematica

t[n_, 1] := CatalanNumber[n-1]; t[n_, k_] := CatalanNumber[n-k+1] * CatalanNumber[k-2]; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2013 *)

Crossrefs

Cf. A000108.

Sequence in context: A204851 A114292 A333724 * A299499 A190215 A190252

Adjacent sequences: A178515 A178516 A178517 * A178519 A178520 A178521

Keyword

nonn,tabl

Author

Emeric Deutsch, May 31 2010

Status

approved