A178515 Triangle read by rows: T(n,k) is the number of involutions of {1,2,...,n} having genus k (see first comment for definition of genus).

1, 2, 0, 4, 0, 0, 9, 1, 0, 0, 21, 5, 0, 0, 0, 51, 25, 0, 0, 0, 0, 127, 105, 0, 0, 0, 0, 0, 323, 420, 21, 0, 0, 0, 0, 0, 835, 1596, 189, 0, 0, 0, 0, 0, 0, 2188, 5880, 1428, 0, 0, 0, 0, 0, 0, 0, 5798, 21120, 8778, 0, 0, 0, 0, 0, 0, 0, 0, 15511, 74415, 48741, 1485, 0, 0, 0, 0, 0, 0, 0, 0, 41835, 258115, 249249, 19305, 0

Offset

1,2

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 A000085(n).

T(n,0)=A001006(n) (the Motzkin numbers).

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..83.

Example

T(4,1)=1 because p=3412 is the only involution of {1,2,3,4} with genus 1. This follows easily from the fact that 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). [Also, for p=3412=(13)(24) we have cp'=2341*3412=4123=(1432) and so g(p)=(1/2)(4+1-2-1)=1.]
Triangle starts:
[ 1]  1,
[ 2]  2, 0,
[ 3]  4, 0, 0,
[ 4]  9, 1, 0, 0,
[ 5]  21, 5, 0, 0, 0,
[ 6]  51, 25, 0, 0, 0, 0,
[ 7]  127, 105, 0, 0, 0, 0, 0,
[ 8]  323, 420, 21, 0, 0, 0, 0, 0,
[ 9]  835, 1596, 189, 0, 0, 0, 0, 0, 0,
[10]  2188, 5880, 1428, 0, 0, 0, 0, 0, 0, 0,
[11]  5798, 21120, 8778, 0, 0, 0, 0, 0, 0, 0, 0,
[12]  15511, 74415, 48741, 1485, 0, 0, 0, 0, 0, 0, 0, 0,
[13]  41835, 258115, 249249, 19305, 0, 0, 0, 0, 0, 0, 0, 0, 0,
[14]  113634, 883883, 1201200, 191763, 0, 0, 0, 0, 0, 0, 0, ...,
[15]  310572, 2994355, 5519514, 1525095, 0, 0, 0, 0, 0, 0, 0, ...,
[16]  853467, 10051860, 24408384, 10667800, 225225, 0, 0, 0, ...,
[17]  2356779, 33479460, 104552448, 67581800, 3828825, 0, 0, ...,
...

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: 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: nrcyc := proc (p) local pc: 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; INV := {}: for i to factorial(n) do if inv(P[i]) = P[i] then INV := `union`(INV, {P[i]}) else end if end do: f[n] := sort(add(t^gen(INV[j]), j = 1 .. nops(INV))): seq(coeff(f[n], t, j), j = 0 .. degree(f[n])); # yields the entries of the specified row n

Crossrefs

Cf. A177267.

Cf. A000085, A001006.

Sequence in context: A249093 A102392 A227311 * A344874 A051517 A289359

Adjacent sequences: A178512 A178513 A178514 * A178516 A178517 A178518

Keyword

nonn,hard,tabl

Author

Emeric Deutsch, May 29 2010

Extensions

Terms beyond row 7 from Joerg Arndt, Nov 01 2012.

Status

approved