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 1-5 of 5 results.

A058094 Number of 321-hexagon-avoiding permutations in S_n, i.e., permutations of 1..n with no submatrix equivalent to 321, 56781234, 46781235, 56718234 or 46718235.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 132, 429, 1426, 4806, 16329, 55740, 190787, 654044, 2244153, 7704047, 26455216, 90860572, 312090478, 1072034764, 3682565575, 12650266243, 43456340025, 149282561256, 512821712570, 1761669869321, 6051779569463, 20789398928496, 71416886375493
Offset: 0

Views

Author

Sara Billey, Dec 03 2000

Keywords

Comments

If y is 321-hexagon avoiding, there are simple explicit formulas for all the Kazhdan-Lusztig polynomials P_{x,y} and the Kazhdan-Lusztig basis element C_y is the product of C_{s_i}'s corresponding to any reduced word for y.

Examples

			Since the Catalan numbers count 321-avoiding permutations in S_n, a(8) = 1430 - 4 = 1426 subtracting the four forbidden hexagon patterns.

Links

Crossrefs

Cf. A000108, A092489, A092490, A092491, A092492, A092493.

Programs

  • Maple
    a[0]:=1: a[1]:=1: a[2]:=2: a[3]:=5: a[4]:=14: a[5]:=42: for n from 5 to 35 do a[n+1]:=6*a[n]-11*a[n-1]+9*a[n-2]-4*a[n-3]-4*a[n-4]+a[n-5] od: seq(a[n], n=0..35);
  • Mathematica
    LinearRecurrence[{6,-11,9,-4,-4,1},{1,2,5,14,42,132},40] (* Harvey P. Dale, Nov 09 2012 *)

Formula

a(n+1) = 6a(n) - 11a(n-1) + 9a(n-2) - 4a(n-3) - 4a(n-4) + a(n-5) for n >= 5.
O.g.f.: 1 -x*(1-4*x+4*x^2-3*x^3-x^4+x^5)/(-1+6*x-11*x^2+9*x^3-4*x^4 -4*x^5 +x^6). - R. J. Mathar, Dec 02 2007

Extensions

More terms from Emeric Deutsch, May 04 2004
a(0) prepended by Alois P. Heinz, Sep 21 2014

A092492 Arises in enumeration of 321-hexagon-avoiding permutations.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 5, 19, 68, 240, 839, 2911, 10054, 34641, 119203, 409893, 1408873, 4841373, 16634350, 57149111, 196333312, 674477710, 2317047808, 7959739375, 27343914410, 93933688630, 322686958885, 1108513737048, 3808031504891
Offset: 1

Views

Author

N. J. A. Sloane, Apr 04 2004

Keywords

Links

Crossrefs

Cf. A058094, A092489, A092490, A092491.

Programs

  • Maple
    b[1]:=1: b[2]:=2: b[3]:=5: b[4]:=14: b[5]:=42: b[6]:=132: for n from 6 to 45 do b[n+1]:=6*b[n]-11*b[n-1]+9*b[n-2]-4*b[n-3]-4*b[n-4]+b[n-5] od: a[1]:=0: a[2]:=0: a[3]:=0: a[4]:=0: a[5]:=0: for n from 6 to 40 do a[n]:=2*b[n-3]-5*b[n-4]+b[n-5] od: seq(a[n],n=1..40); # Emeric Deutsch, Jun 08 2004
  • Mathematica
    LinearRecurrence[{6,-11,9,-4,-4,1},{0,0,0,0,0,1,5},30] (* Harvey P. Dale, Jan 31 2025 *)
  • PARI
    concat([0,0,0,0,0], Vec(x^6*(1 - x) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6) + O(x^30))) \\ Colin Barker, Aug 21 2019

Formula

a(n) = 2*A058094(n-3) - 5*A058094(n-4) + A058094(n-5) for n >= 6. - Emeric Deutsch, Jun 08 2004
From Colin Barker, Aug 21 2019: (Start)
G.f.: x^6*(1 - x) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6).
a(n) = 6*a(n-1) - 11*a(n-2) + 9*a(n-3) - 4*a(n-4) - 4*a(n-5) + a(n-6) for n>7.
(End)

Extensions

More terms from Emeric Deutsch, Jun 08 2004

A092490 a(n) = A058094(n) - 3*A058094(n-1) + A058094(n-2) for n >=4.

Original entry on oeis.org

0, 0, 0, 1, 5, 20, 75, 271, 957, 3337, 11559, 39896, 137423, 472808, 1625632, 5587228, 19198971, 65963978, 226623902, 778551761, 2674604282, 9188106871, 31563807424, 108430368827, 372487292867, 1279591674070, 4395730089428
Offset: 1

Views

Author

N. J. A. Sloane, Apr 04 2004

Keywords

Comments

A recurrence relation follows in a straightforward manner from the above formula and the recurrence relation for A058094.

Links

Crossrefs

Cf. A058094, A092489, A092491, A092492.
Cf. A058094.

Programs

  • Maple
    b[1]:=1:b[2]:=2:b[3]:=5:b[4]:=14:b[5]:=42:b[6]:=132: for n from 6 to 32 do b[n+1]:=6*b[n]-11*b[n-1]+9*b[n-2]-4*b[n-3]-4*b[n-4]+b[n-5] od:a[1]:=0:a[2]:=0:a[3]:=0:for n from 4 to 32 do a[n]:=b[n]-3*b[n-1]+b[n-2] od: seq(a[n],n=1..32); # Emeric Deutsch, Apr 12 2005
  • PARI
    concat([0,0,0], Vec(x^4*(1 - x + x^2 + x^3) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6) + O(x^30))) \\ Colin Barker, Aug 21 2019

Formula

G.f.: x^4*(1 - x + x^2 + x^3) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6). - R. J. Mathar, Dec 02 2007
a(n) = 6*a(n-1) - 11*a(n-2) + 9*a(n-3) - 4*a(n-4) - 4*a(n-5) + a(n-6) for n>7. - Colin Barker, Aug 21 2019

Extensions

Edited by Emeric Deutsch, Apr 12 2005

A092491 a(n) = 2*A058094(n-2) - 5*A058094(n-3) + A058094(n-4) + a(n-1) for n >=4.

Original entry on oeis.org

0, 0, 0, 0, 1, 6, 25, 93, 333, 1172, 4083, 14137, 48778, 167981, 577874, 1986747, 6828120, 23462470, 80611581, 276944893, 951422603, 3268470411, 11228209786, 38572124196, 132505812826, 455192771711, 1563706508759, 5371738013650
Offset: 1

Views

Author

N. J. A. Sloane, Apr 04 2004

Keywords

Comments

A recurrence relation follows in a straightforward manner from the above formula and the recurrence relation for A058094.

Links

Crossrefs

Cf. A058094, A092489, A092490, A092492.

Programs

  • Maple
    b[1]:=1:b[2]:=2:b[3]:=5:b[4]:=14:b[5]:=42:b[6]:=132: for n from 6 to 34 do b[n+1]:=6*b[n]-11*b[n-1]+9*b[n-2]-4*b[n-3]-4*b[n-4]+b[n-5] od: a[1]:=0:a[2]:=0:a[3]:=0:a[4]:=0: for n from 5 to 34 do a[n]:=2*b[n-2]-5*b[n-3]+b[n-4]+a[n-1] od: seq(a[n],n=1..34); # Emeric Deutsch, Apr 12 2005
  • Mathematica
    LinearRecurrence[{6, -11, 9, -4, -4, 1}, {0, 0, 0, 0, 1, 6}, 40] (* Vincenzo Librandi, Aug 15 2017 *)
  • PARI
    concat([0,0,0,0], Vec(x^5 / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6) + O(x^30))) \\ Colin Barker, Aug 21 2019

Formula

From Colin Barker, Aug 21 2019: (Start)
G.f.: x^5 / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6).
a(n) = 6*a(n-1) - 11*a(n-2) + 9*a(n-3) - 4*a(n-4) - 4*a(n-5) + a(n-6) for n>6.
(End)

Extensions

Edited by Emeric Deutsch, Apr 12 2005

A092493 a(n) = 4a(n-1) - 4a(n-2) + 3a(n-3) + a(n-4) - a(n-5).

Original entry on oeis.org

1, 2, 5, 14, 42, 128, 389, 1179, 3572, 10825, 32810, 99446, 301412, 913547, 2768863, 8392136, 25435699, 77092976, 233660832, 708201794, 2146486339, 6505777953, 19718339694, 59764246943, 181139247400, 549014312524, 1664005563066
Offset: 1

Views

Author

N. J. A. Sloane, Apr 04 2004

Keywords

Comments

Arises in enumeration of certain pattern-avoiding permutations.

Links

Crossrefs

Cf. A058094, A092489-A092492.

Programs

  • Maple
    a[1]:=1: a[2]:=2: a[3]:=5: a[4]:=14: a[5]:=42: for n from 6 to 32 do a[n]:=4*a[n-1]-4*a[n-2]+3*a[n-3]+a[n-4]-a[n-5] od: seq(a[j],j=1..32); # Emeric Deutsch, Apr 12 2005
  • Mathematica
    LinearRecurrence[{4,-4,3,1,-1},{1,2,5,14,42},40] (* Harvey P. Dale, Jul 14 2024 *)

Formula

G.f.: x*(1 - 2*x + x^2 - x^3 - x^4)/(1 - 4*x + 4*x^2 - 3*x^3 - x^4 + x^5). [Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009; corrected by R. J. Mathar, Sep 16 2009]

Extensions

Edited by Emeric Deutsch, Apr 12 2005
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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