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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A370507 T(n,k) is the number permutations p of [n] that are k-dist-increasing but not j-dist-increasing for any j=0 and p(i)=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 5, 7, 11, 0, 1, 9, 22, 33, 55, 0, 1, 19, 77, 112, 192, 319, 0, 1, 34, 189, 480, 788, 1315, 2233, 0, 1, 69, 526, 2187, 3500, 5987, 10409, 17641, 0, 1, 125, 1625, 6811, 18273, 30568, 53791, 92917, 158769, 0, 1, 251, 4111, 23507, 101424, 167480, 299769, 528253, 925337, 1578667
Offset: 0

Views

Author

Alois P. Heinz, Feb 20 2024

Keywords

Examples

			T(4,1) = 1: 1234.
T(4,2) = 5: 1243, 1324, 2134, 2143, 3142.
T(4,3) = 7: 1342, 1423, 1432, 2314, 2413, 3124, 3214.
T(4,4) = 11: 2341, 2431, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   2,    3;
  0, 1,   5,    7,   11;
  0, 1,   9,   22,   33,    55;
  0, 1,  19,   77,  112,   192,   319;
  0, 1,  34,  189,  480,   788,  1315,  2233;
  0, 1,  69,  526, 2187,  3500,  5987, 10409, 17641;
  0, 1, 125, 1625, 6811, 18273, 30568, 53791, 92917, 158769;
  ...

Links

Crossrefs

Columns k=0-2 give: A000007, A057427, A014495.
Main diagonal gives A370514, also A370506(n,1) for n>=1.
Row sums give A000142.
Cf. A370505.

Programs

  • Maple
    q:= proc(l, k) local i; for i from 1 to nops(l)-k do
      if l[i]>=l[i+k] then return 0 fi od; 1
    end:
m:= proc(l) local k;
      for k from 0 to nops(l) do if q(l, k)>0 then return k fi od
    end:
b:= proc(n) b(n):= add(x^m(l), l=combinat[permute](n)) end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n, k), k=0..n), n=0..8);
  • Mathematica
    q[l_, k_] := Module[{i}, For[i = 1, i <= Length[l] - k, i++, If[l[[i]] >= l[[i + k]], Return [0]]]; 1];
m[l_] := Module[{k}, For[k = 0, k <= Length[l], k++, If[q[l, k] > 0, Return[k]]]];
b[n_] := Sum[x^m[l], {l, Permutations[Range@n]}];
T[n_, k_] := Coefficient[b[n], x, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 8}] // Flatten (* Jean-François Alcover, Feb 29 2024, after Alois P. Heinz *)

A370505 T(n,k) is the difference between the number of k-dist-increasing and (k-1)-dist-increasing permutations of [n], where p is k-dist-increasing if k>=0 and p(i)=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 5, 6, 12, 0, 1, 9, 20, 30, 60, 0, 1, 19, 70, 90, 180, 360, 0, 1, 34, 175, 420, 630, 1260, 2520, 0, 1, 69, 490, 1960, 2520, 5040, 10080, 20160, 0, 1, 125, 1554, 5880, 15120, 22680, 45360, 90720, 181440, 0, 1, 251, 3948, 21000, 88200, 113400, 226800, 453600, 907200, 1814400
Offset: 0

Views

Author

Alois P. Heinz, Feb 20 2024

Keywords

Examples

			T(0,0) = 1: (only) the empty permutation is 0-dist-increasing.
T(4,2) = 5 = 6 - 1 = |{1234, 1243, 1324, 2134, 2143, 3142}| - |{1234}|.
Permutation 3142 is 2-dist-increasing and 4-dist-increasing but not 3-dist-increasing.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   2,    3;
  0, 1,   5,    6,   12;
  0, 1,   9,   20,   30,    60;
  0, 1,  19,   70,   90,   180,   360;
  0, 1,  34,  175,  420,   630,  1260,  2520;
  0, 1,  69,  490, 1960,  2520,  5040, 10080, 20160;
  0, 1, 125, 1554, 5880, 15120, 22680, 45360, 90720, 181440;
  ...

Links

Crossrefs

Columns k=0-2 give: A000007, A057427, A014495.
Row sums give A000142.
Main diagonal gives A001710.
T(2n,n+1) gives A000680 for n>=1.
T(2n,n) gives A370576.
Cf. A248686, A248687, A370506, A370507.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k<1,
     `if`(n=k, 1, 0), n!/mul(iquo(n+i, k)!, i=0..k-1))
    end:
T:= (n, k)-> b(n, k)-b(n, k-1):
seq(seq(T(n, k), k=0..n), n=0..10);

Formula

T(n,k) = A248686(n,k) - A248686(n,k-1) for k>=2.
Sum_{k=0..n} (1+n-k) * T(n,k) = A248687(n) for n>=1.

A370514 Number of permutations p of [n] such that for each distance d in [n-1] there is at least one index i in [n-d] with p(i)>p(i+d).

Original entry on oeis.org

1, 1, 1, 3, 11, 55, 319, 2233, 17641, 158769, 1578667, 17365337, 207865289, 2702248757, 37786779669, 566801695035, 9063808803203, 154084749654451
Offset: 0

Views

Author

Alois P. Heinz, Feb 20 2024

Keywords

Examples

			a(0) = 1: the empty permutation.
a(1) = 1: 1.
a(2) = 1: 21.
a(3) = 3: 231, 312, 321.
a(4) = 11: 2341, 2431, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321.
a(5) = 55: 23451, 23541, 24351, 24531, ..., 54213, 54231, 54312, 54321.
a(6) = 319: 234561, 234651, 235461, 235641, ..., 654213, 654231, 654312, 654321.

Links

Crossrefs

Main diagonal of A370507.
Column k=1 of A370506 (for n>=1).
Cf. A008302.

Formula

a(n) = A370507(n,n).
a(n) = A370506(n,1) for n>=1.

Extensions

a(14)-a(16) from Martin Ehrenstein, Feb 22 2024
a(17) from Alois P. Heinz, Feb 22 2024
Showing 1-3 of 3 results.

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