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A178516
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Triangle read by rows: T(n,k) is the number of up-down permutations of {1,2,...,n} having genus k (see first comment for definition of genus).
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4
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1, 1, 0, 2, 0, 0, 2, 3, 0, 0, 6, 10, 0, 0, 0, 6, 38, 17, 0, 0, 0, 22, 142, 104, 4, 0, 0, 0, 22, 351, 778, 234, 0, 0, 0, 0, 90, 1419, 4086, 2235, 106, 0, 0, 0, 0, 90, 2856, 17402, 24357, 5816, 0, 0, 0, 0, 0, 394, 12208, 87434, 171305, 78705, 3746, 0, 0, 0, 0, 0, 394, 21676, 278062, 1053425, 1120648, 228560, 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 A000111(n) (Euler or up-down numbers).
Apparently, row n contains ceiling(n/2) nonzero entries.
T(2n-1,0) = T(2n,0) = A006318(n-1) (the large Schroeder numbers).
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LINKS
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EXAMPLE
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T(4,0)=2. 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), it follows that the up-down permutations 2314 = (123)(4) and 1324 = (1)(23)(4) have genus 0, while 2413 = (1243), 3412 = (13)(24), and 1423 = (1)(243) do not.
Triangle starts:
[ 1] 1,
[ 2] 1, 0,
[ 3] 2, 0, 0,
[ 4] 2, 3, 0, 0,
[ 5] 6, 10, 0, 0, 0,
[ 6] 6, 38, 17, 0, 0, 0,
[ 7] 22, 142, 104, 4, 0, 0, 0,
[ 8] 22, 351, 778, 234, 0, 0, 0, 0,
[ 9] 90, 1419, 4086, 2235, 106, 0, 0, 0, 0,
[10] 90, 2856, 17402, 24357, 5816, 0, 0, 0, 0, 0,
[11] 394, 12208, 87434, 171305, 78705, 3746, 0, 0, 0, 0, 0,
[12] 394, 21676, 278062, 1053425, 1120648, 228560, 0, 0, 0, 0, 0, 0,
...
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MAPLE
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n := 7: with(combinat): descents := proc (p) local A, i: A := {}: for i to nops(p)-1 do if p[i+1] < p[i] then A := `union`(A, {i}) else end if end do: A end proc; UD := proc (n) local ud, P, j: ud := {}: P := permute(n): for j to factorial(n) do if descents(P[j]) = {seq(2*k, k = 1 .. ceil((1/2)*n)-1)} then ud := `union`(ud, {P[j]}) else end if end do: ud end proc; 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: nrcyc := proc (p) local nrfp, pc: 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:
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: f[n] := sort(add(t^gen(UD(n)[j]), j = 1 .. nops(UD(n)))): seq(coeff(f[n], t, j), j = 0 .. ceil((1/2)*n)-1); # yields the entries in the specified row n
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CROSSREFS
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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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