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A178517
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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).
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3
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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
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OFFSET
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1,4
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COMMENTS
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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).
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REFERENCES
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S. Dulucq and R. Simion, Combinatorial statistics on alternating permutations, J. Algebraic Combinatorics, 8 (1998), 169-191.
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LINKS
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EXAMPLE
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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, ...,
...
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MAPLE
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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
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CROSSREFS
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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).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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