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A303285
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Number of permutations p of [2n] such that the sequence of ascents and descents of p0 forms a Dyck path.
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9
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1, 1, 8, 172, 7296, 518324, 55717312, 8460090160, 1726791794432, 456440969661508, 151770739970889792, 62022635037246022000, 30564038464166725328768, 17876875858414492985045712, 12245573879235563308351042496, 9711714975145772145881269175104
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OFFSET
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0,3
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COMMENTS
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Here p is a permutation of 1,2,3,...,2n, and p0 refers to the string p followed by 0.
Also the number of permutations p of [2n] such that the sequence of ascents and descents of 0p forms a Dyck path. a(2) = 8: 1432, 2143, 2431, 3142, 3241, 3421, 4132, 4231.
Also the number of permutations p of [2n] that are of odd order and whose M statistic (as defined in the Spiro paper) is equal to n-1. - Sam Spiro, Nov 01 2018
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LINKS
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FORMULA
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a(n) ~ c * 2^(2*n) * n^(2*n - 1) / exp(2*n), where c = 8.838022110416151362523442920999767406145711133564692... - Vaclav Kotesovec, May 22 2018
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EXAMPLE
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a(0) = 1: the empty permutation.
a(1) = 1: 12.
a(2) = 8: 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.
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MAPLE
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b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
`if`(t>0, add(b(u-j, o+j-1, t-1), j=1..u), 0)+
`if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
end:
a:= n-> b(0, 2*n, 0):
seq(a(n), n=0..20);
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MATHEMATICA
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b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t > 0, Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] + If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
a[n_] := b[0, 2n, 0];
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PROG
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b(n)={if(n==0, 1, 2*sum(k=0, n, (-1)^k*binomial(2*n+1, k)*(n-k+1)^(2*n))); }
a(n)={if(n==0, 1, sum(k=1, n, binomial(2*n, 2*k-1)*b(k-1)*b(n-k))/2); } \\ Andrew Howroyd, Nov 01 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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