A262131 -id:A262131 - cp's OEIS frontend

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.

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

Offset: 0

Alois P. Heinz, Sep 11 2015

			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, ...

Alois P. Heinz, Antidiagonals n = 0..100, flattened

Columns k=1-10 give: A000111, A262126, A262128, A262129, A262130, A262131, A262132, A262133, A262134, A262135.
Main diagonal gives A000246.
Cf. A258829, A262125.

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);

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 *)

A(n,k) = Sum_{i=0..k} A262125(n,i).

A320980 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of six.

Original entry on oeis.org

0, 1, 7, 514, 4189, 154261, 1477381, 44169020, 493190771, 13821362271, 177705152975, 4949371839867, 72355179873697, 2058206624313873, 33818827542140211, 995975339452380880, 18206096557050382759, 558929622195992201388, 11264684856271486133087

Offset: 6

Alois P. Heinz, Oct 25 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..452

Column k=6 of A262125.

Cf. A262130, A262131.

b:= proc(u, o, c) option remember; `if`(c<0 or c>6, 0, `if`(u+o=0,
       x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..6))(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-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 6):
seq(a(n), n=6..30);

a(n) = A262131(n) - A262130(n).

A320981 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of seven.

Original entry on oeis.org

0, 1, 8, 1031, 9379, 486299, 5084162, 196352061, 2352460536, 81070646577, 1103813259377, 36592927821767, 560827842703887, 18549898652794829, 317078625531545481, 10667284286197389079, 201655098112826170280, 6973904694490809821089, 144705018721890264334923

Offset: 7

Alois P. Heinz, Oct 25 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..452

Column k=7 of A262125.

Cf. A262131, A262132.

b:= proc(u, o, c) option remember; `if`(c<0 or c>7, 0, `if`(u+o=0,
       x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..7))(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-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 7):
seq(a(n), n=7..30);

a(n) = A262132(n) - A262131(n).

cp's OEIS Frontend

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.

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A320980 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of six.

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Maple

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A320981 Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of seven.

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Author

Keywords

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Crossrefs

Programs

Maple

Formula