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

Author

Emeric Deutsch, May 29 2010

Keywords

• nonn
• hard
• 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 A000085(n).

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

Examples

			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, ...,
...
		

References

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

Crossrefs

Cf. A177267.

Cf. A000085, A001006.

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: 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

Extensions

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