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 A262124 Number A(n,k) of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals. 16
 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1, 3, 5, 0, 1, 1, 1, 3, 8, 16, 0, 1, 1, 1, 3, 9, 40, 61, 0, 1, 1, 1, 3, 9, 44, 162, 272, 0, 1, 1, 1, 3, 9, 45, 219, 1134, 1385, 0, 1, 1, 1, 3, 9, 45, 224, 1445, 6128, 7936, 0, 1, 1, 1, 3, 9, 45, 225, 1568, 9985, 55152, 50521, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,14 LINKS Alois P. Heinz, Antidiagonals n = 0..100, flattened FORMULA A(n,k) = Sum_{i=0..k} A262125(n,i). EXAMPLE p = 1423 is counted by T(4,1) because the up-down signature of p = 1423 is 1,-1,1 with partial sums 1,0,1. q = 1432 is not counted by any T(4,k) because the up-down signature of q = 1432 is 1,-1,-1 with partial sums 1,0,-1. A(4,1) = 5: 1324, 1423, 2314, 2413, 3412. A(4,2) = 8: 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412. A(4,3) = 9: 1234, 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412. Square array A(n,k) begins:   1,   1,    1,    1,    1,    1,    1,    1, ...   1,   1,    1,    1,    1,    1,    1,    1, ...   0,   1,    1,    1,    1,    1,    1,    1, ...   0,   2,    3,    3,    3,    3,    3,    3, ...   0,   5,    8,    9,    9,    9,    9,    9, ...   0,  16,   40,   44,   45,   45,   45,   45, ...   0,  61,  162,  219,  224,  225,  225,  225, ...   0, 272, 1134, 1445, 1568, 1574, 1575, 1575, ... 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: A:= (n, k)-> `if`(n=0, 1, (p-> add(coeff(p, x, i), i=0..min(n, k))               )(add(b(j-1, n-j, 0), j=1..n))): seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA b[u_, o_, c_] := b[u, o, c] = If[c<0, 0, If[u+o == 0, x^c, Function[p, Sum[ Coefficient[p, x, i]*x^Max[i, c], {i, 0, Exponent[p, 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}]]]]; A[n_, k_] := If[n==0, 1, Function[p, Sum[Coefficient[p, x, i], {i, 0, Min[n, k]}]][ Sum[b[j-1, n-j, 0], {j, 1, n}]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *) CROSSREFS Columns k=1-10 give: A000111, A262126, A262128, A262129, A262130, A262131, A262132, A262133, A262134, A262135. Main diagonal gives A000246. Cf. A258829, A262125. Sequence in context: A187596 A263863 A134655 * A199954 A219987 A077614 Adjacent sequences:  A262121 A262122 A262123 * A262125 A262126 A262127 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 11 2015 STATUS approved

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Last modified January 16 23:44 EST 2019. Contains 319206 sequences. (Running on oeis4.)