cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A344216 a(n) = n!*((n+1)/2 + 2*Sum_{k=2..n-1}(n-k)/(k+1)).

Original entry on oeis.org

1, 3, 16, 104, 768, 6336, 57888, 581472, 6379200, 75977280, 977045760, 13499930880, 199537067520, 3142504512000, 52546707763200, 929908914278400, 17366044153651200, 341336836618444800, 7044417438363648000
Offset: 1

Views

Author

F. Chapoton, May 13 2021

Keywords

Comments

Conjecture: a(n) is the number of linear intervals in the weak order on the symmetric group S_n. An interval is linear if it is isomorphic to a total order. The conjecture has been checked up to a(7) = 57888.

Examples

			For S_3, among the 17 intervals in the hexagon-shaped lattice, only the full lattice is not linear.

Links

Crossrefs

Cf. A344136, A344191, A344228 for similar sequences.
Cf. A007767 for all intervals in the weak order on S_n.
Cf. A001008, A002805.

Programs

  • Maple
    a := n -> (1/2)*n!*(4*(n + 1)*harmonic(n) - 9*n + 3):
# Or:
egf := (3 - 8*x - 4*ln(1 - x))/(2*(x - 1)^2):
ser := series(egf, x, 24): a := n -> n!*coeff(ser, x, n):
seq(a(n), n=1..19); # Peter Luschny, May 13 2021
  • Mathematica
    Join[{1}, RecurrenceTable[{(n - 3) a[n] == (2 n^2 - 5 n - 1) a[n - 1] - (n^3 - 3 n^2 + 2 n) a[n - 2], a[2] == 3, a[3] == 16}, a, {n, 2, 19}]] (* Peter Luschny, May 13 2021 *)
  • PARI
    a(n) = n!*((n+1)/2+2*sum(k=2, n-1, (n-k)/(k+1))); \\ Michel Marcus, May 13 2021
  • Sage
    def a(n):
    return factorial(n)*((n+1)/2+2*sum((n-k)/(k+1) for k in range(2, n)))

Formula

From Peter Luschny, May 13 2021: (Start)
a(n) = (1/2) * n! * (4 * (n + 1) * H(n) - 9*n + 3), where H(n) are the harmonic numbers H(n) = A001008(n)/A002805(n).
a(n) = n! * [x^n] (3 - 8*x - 4*log(1 - x))/(2*(x - 1)^2).
a(n) = ((2*n^2 - 5*n - 1)*a(n-1) - (n^3 - 3*n^2 + 2*n)*a(n-2))/(n - 3) for n >= 4. (End)

A263754 Triangle read by rows: T(n,k) (n>=0, 1<=k<=n!) is the number of permutations pi of n such that there are k permutations <= pi in the left weak order.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 0, 0, 1, 1, 3, 4, 3, 2, 3, 0, 4, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 4, 6, 7, 6, 9, 4, 10, 4, 8, 2, 8, 0, 4, 8, 2, 0, 4, 0, 9, 0, 0, 0, 2, 4, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Christian Stump, Oct 19 2015

Keywords

Comments

Row sums give A000142.

Examples

			Triangle begins:
  1;
  1;
  1,1;
  1,2,2,0,0,1;
  1,3,4,3,2,3,0,4,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1;
  ...

Links

Crossrefs

Cf. A000142, A007767.

Formula

Sum_{k=1..n!} k * T(n,k) = A007767(n). - Alois P. Heinz, Jun 06 2016

Extensions

Row n=0 prepended by Alois P. Heinz, Jun 06 2016

A190291 Number of intervals in the weak (Bruhat) order of the symmetric group S_n that are distributive lattices.

Original entry on oeis.org

1, 3, 16, 124, 1262, 15898, 238572, 4152172
Offset: 1

Views

Author

Richard Stanley, May 07 2011

Keywords

Comments

The intervals [u,v] in the weak order that are distributive lattices are characterized by Stembridge. They are the intervals such that u^{-1}.v is fully commutative, i.e., avoids the pattern 321.

Examples

			Example: for n=3 there are six 1-element intervals, six 2-element intervals, and four intervals that are 3-element chains, for a total of 16.

Links

Crossrefs

Cf. A007767.
Showing 1-3 of 3 results.

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