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A262125
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Number T(n,k) of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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15
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1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 5, 3, 1, 0, 0, 16, 24, 4, 1, 0, 0, 61, 101, 57, 5, 1, 0, 0, 272, 862, 311, 123, 6, 1, 0, 0, 1385, 4743, 3857, 778, 254, 7, 1, 0, 0, 7936, 47216, 27589, 14126, 1835, 514, 8, 1, 0, 0, 50521, 322039, 355751, 111811, 47673, 4189, 1031, 9, 1, 0
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OFFSET
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0,8
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LINKS
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Alois P. Heinz, Rows n = 0..100, flattened
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FORMULA
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T(n,k) = A262124(n,k) - A262124(n,k-1) for k>0, T(n,0) = A262124(n,0).
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EXAMPLE
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T(4,1) = 5: 1324, 1423, 2314, 2413, 3412.
T(4,2) = 3: 1243, 1342, 2341.
T(4,3) = 1: 1234.
Triangle T(n,k) begins:
1;
1, 0;
0, 1, 0;
0, 2, 1, 0;
0, 5, 3, 1, 0;
0, 16, 24, 4, 1, 0;
0, 61, 101, 57, 5, 1, 0;
0, 272, 862, 311, 123, 6, 1, 0;
0, 1385, 4743, 3857, 778, 254, 7, 1, 0;
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MAPLE
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b:= proc(u, o, c) option remember; `if`(c<0, 0, `if`(u+o=0, x^c,
(p-> add(coeff(p, x, i)*x^max(i, c), i=0..degree(p)))(add(
b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
end:
T:= n-> `if`(n=0, 1, (p-> seq(coeff(p, x, i), i=0..n)
)(add(b(j-1, n-j, 0), j=1..n))):
seq(T(n), n=0..10);
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MATHEMATICA
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b[u_, o_, c_] := b[u, o, c] = If[c<0, 0, If[u+o==0, x^c, Sum[Coefficient[ #, x, i]*x^Max[i, c], {i, 0, Exponent[#, x]}]]& @ Sum[b[u-j, o-1+j, c-1], {j, 1, u}] + Sum[b[u+j-1, o-j, c+1], {j, 1, o}]];
T[n_] := If[n==0, {1}, Table[Coefficient[#, x, i], {i, 0, n}]]& @ Sum[b[j-1, n-j, 0], {j, 1, n}];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, Jan 19 2020, after Alois P. Heinz *)
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CROSSREFS
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Columns k=1-10 give: A000111 (for n>1), A320976, A320977, A320978, A320979, A320980, A320981, A320982, A320983, A320984.
Row sums give A000246.
T(2n,n) gives A262127.
Cf. A258829, A262124.
Sequence in context: A079508 A057150 A185663 * A105868 A267163 A265163
Adjacent sequences: A262122 A262123 A262124 * A262126 A262127 A262128
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz, Sep 11 2015
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STATUS
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approved
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