login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A178517 Triangle read by rows: T(n,k) is the number of non-derangement permutations of {1,2,...,n} having genus k (see first comment for definition of genus). 3
1, 1, 0, 4, 0, 0, 11, 4, 0, 0, 36, 40, 0, 0, 0, 117, 290, 48, 0, 0, 0, 393, 1785, 1008, 0, 0, 0, 0, 1339, 9996, 12712, 1440, 0, 0, 0, 0, 4630, 52584, 123858, 48312, 0, 0, 0, 0, 0, 16193, 264720, 1027446, 904840, 80640, 0, 0, 0, 0, 0, 57201, 1290135, 7627158, 12449800, 3807936, 0 (list; table; graph; refs; listen; history; text; internal format)
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.

The sum of the entries in row n is A002467(n).

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

T(n,0) = A106640(n-1) .

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..61.

EXAMPLE

T(3,0)=4 because all non-derangements of {1,2,3}, namely 123=(1)(2)(3), 132=(1)(23), 213=(12)(3), and 321=(13)(2) have genus 0. 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).

Triangle starts:

[ 1]  1,

[ 2]  1, 0,

[ 3]  4, 0, 0,

[ 4]  11, 4, 0, 0,

[ 5]  36, 40, 0, 0, 0,

[ 6]  117, 290, 48, 0, 0, 0,

[ 7]  393, 1785, 1008, 0, 0, 0, 0,

[ 8]  1339, 9996, 12712, 1440, 0, 0, 0, 0,

[ 9]  4630, 52584, 123858, 48312, 0, 0, 0, 0, 0,

[10]  16193, 264720, 1027446, 904840, 80640, 0, 0, 0, 0, 0,

[11]  57201, 1290135, 7627158, 12449800, 3807936, 0, 0, 0, 0, 0, 0,

[12]  203799, 6133930, 52188774, 140356480, 96646176, 7257600, 0, ...,

[13]  731602, 28603718, 335517468, 1373691176, 1749377344, 448306560, 0, ...,

...

MAPLE

n := 7: 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: NDER := {}: for i to factorial(n) do if nrfp(P[i]) > 0 then NDER := `union`(NDER, {P[i]}) else end if end do: f[n] := sort(add(t^gen(NDER[j]), j = 1 .. nops(NDER))): seq(coeff(f[n], t, j), j = 0 .. floor((1/2)*n)-1); # yields the entries in the specified row n

CROSSREFS

Cf. A177267 (genus of all permutations).

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

Cf. A002467, A106640.

Sequence in context: A271910 A249346 A035539 * A049207 A092219 A262227

Adjacent sequences:  A178514 A178515 A178516 * A178518 A178519 A178520

KEYWORD

nonn,hard,tabl

AUTHOR

Emeric Deutsch, May 30 2010

EXTENSIONS

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

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 18 17:56 EST 2017. Contains 294894 sequences.