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 A262125 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. 15
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS Alois P. Heinz, Rows n = 0..100, flattened FORMULA T(n,k) = A262124(n,k) - A262124(n,k-1) for k>0, T(n,0) = A262124(n,0). EXAMPLE 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; MAPLE 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); MATHEMATICA 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 *) CROSSREFS 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 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 11 2015 STATUS approved

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Last modified May 18 22:37 EDT 2022. Contains 353826 sequences. (Running on oeis4.)