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A217323 Number of self-inverse permutations in S_n with longest increasing subsequence of length 3. 3
1, 3, 11, 31, 92, 253, 709, 1936, 5336, 14587, 40119, 110202, 304137, 840597, 2332469, 6487762, 18106906, 50667263, 142194843, 400057791, 1128408337, 3190023641, 9038202201, 25659417876, 72987714502, 207983161609, 593665226069, 1697230353691, 4859461136196 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,2
COMMENTS
Also the number of Young tableaux with n cells and 3 rows.
LINKS
FORMULA
a(n) = A182172(n,3) - A182172(n,2) = A001006(n) - A001405(n).
EXAMPLE
a(3) = 1: 123.
a(4) = 3: 1243, 1324, 2134.
a(5) = 11: 12543, 13254, 14325, 14523, 15342, 21354, 21435, 32145, 34125, 42315, 52341.
MAPLE
a:= proc(n) option remember; `if`(n<3, 0, `if`(n=3, 1,
((n+1)*(6*n^3-5*n^2-7*n-24)*a(n-1)
+n*(n-1)*(21*n^2-28*n-109)*a(n-2)
-2*(n-1)*(n-2)*(12*n^2+11*n-3)*a(n-3)
-12*(3*n+5)*(n-1)*(n-2)*(n-3)*a(n-4))/
((n-3)*(3*n+2)*(n+2)*(n+1))))
end:
seq(a(n), n=3..40);
MATHEMATICA
MotzkinNumber = DifferenceRoot[Function[{y, n}, {(-3n-3)*y[n] + (-2n-5)*y[n+1] + (n+4)*y[n+2] == 0, y[0] == 1, y[1] == 1}]];
a[n_] := MotzkinNumber[n] - Binomial[n, Quotient[n, 2]];
Table[a[n], {n, 3, 40}] (* Jean-François Alcover, Oct 27 2021, from 2nd formula *)
CROSSREFS
Column k=3 of A047884.
Sequence in context: A369399 A108898 A291887 * A263781 A119901 A106934
KEYWORD
nonn,easy
AUTHOR
Alois P. Heinz, Sep 30 2012
STATUS
approved

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Last modified May 26 16:43 EDT 2024. Contains 372840 sequences. (Running on oeis4.)