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.

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A230349 Number of permutations of [2n+9] in which the longest increasing run has length n+9.

Original entry on oeis.org

1, 20, 417, 9690, 253776, 7465176, 244906200, 8891411760, 354610872000, 15432114297600, 728406536457600, 37090538241120000, 2027740775284224000, 118512161081233920000, 7376476698319125196800, 487273386402209523916800, 34055074238462266429440000
Offset: 0

Views

Author

Alois P. Heinz, Oct 16 2013

Keywords

Comments

Also the number of ascending runs of length n+9 in the permutations of [2n+9].

Links

Crossrefs

A diagonal of A008304, A122843.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1+19*n, 2*(2*n+9)*(n+4)*
      (n^3+20*n^2+110*n+109)*a(n-1)/((n+11)*(n^3+17*n^2+73*n+18)))
    end:
seq(a(n), n=0..25);

Formula

a(n) = (n^3+20*n^2+110*n+109)*(2*n+9)!/(n+11)! for n>0, a(0) = 1.
a(n) = A008304(2*n+9,n+9) = A122843(2*n+9,n+9).

A230350 Number of permutations of [2n+10] in which the longest increasing run has length n+10.

Original entry on oeis.org

1, 22, 491, 12032, 328950, 10027440, 339006360, 12628788480, 515033719200, 22855760928000, 1097589192336000, 56754471481344000, 3145763658989952000, 186150029203211673600, 11717355323144959488000, 781981263963810054144000, 55165533654753963657216000
Offset: 0

Views

Author

Alois P. Heinz, Oct 16 2013

Keywords

Comments

Also the number of ascending runs of length n+10 in the permutations of [2n+10].

Links

Crossrefs

A diagonal of A008304, A122843.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1+21*n, 2*(n+5)*(2*n+9)*
      (n^3+22*n^2+132*n+131)*a(n-1)/((n+12)*(n^3+19*n^2+91*n+20)))
    end:
seq(a(n), n=0..25);

Formula

a(n) = (n^3+22*n^2+132*n+131)*(2*n+10)!/(n+12)! for n>0, a(0) = 1.
a(n) = A008304(2*n+10,n+10) = A122843(2*n+10,n+10).

A122844 Triangle read by rows: T[n,k] = the number of ascending runs of length at least k in the permutations of [n] for k <= n.

Original entry on oeis.org

1, 3, 1, 12, 5, 1, 60, 28, 7, 1, 360, 180, 50, 9, 1, 2520, 1320, 390, 78, 11, 1, 20160, 10920, 3360, 714, 112, 13, 1, 181440, 100800, 31920, 7056, 1176, 152, 15, 1, 1814400, 1028160, 332640, 75600, 13104, 1800, 198, 17, 1
Offset: 1

Views

Author

David Scambler, Sep 13 2006

Keywords

Comments

Column T[n,1] is essentially A001710 - all ascending runs in permutations of [n] Column T[n,2] is A006157 - ascending runs of length at least 2 in permutations of [n] Column T[n,3] is A005460 - ascending runs of length at least 3 in permutations of [n]

Examples

			1
3 1
12 5 1 ; there are 5 ascending runs of length at least 2 in the permutations of [3], namely 13 in 132 and in 213, 23 in 231, 12 in 312, 123 in 123. T[3,2] = 5.

Crossrefs

Cf. A122844, A001710, A006157, A005460.

Formula

T[n,k] = n![k(n-k+1)+1]/(k+1)! for 0A122843(n,j) (partial row sums of A122843)

A229002 Total sum of the n-th powers of lengths of ascending runs in all permutations of [n].

Original entry on oeis.org

0, 1, 6, 66, 1110, 25620, 765506, 28544040, 1293790126, 69860663220, 4422094936842, 323816329558128, 27127369640967206, 2575241880204602700, 274755427187762475922, 32708158728316937527944, 4316964980670466411606110, 628131523035069583394938980
Offset: 0

Views

Author

Alois P. Heinz, Sep 10 2013

Keywords

Links

Crossrefs

Main diagonal of A229001.

Programs

  • Maple
    a:= n-> add(`if`(n=k, 1, n!/(k+1)!*(k*(n-k+1)+1
        -((k+1)*(n-k)+1)/(k+2)))*k^n, k=1..n):
seq(a(n), n=1..20);

Formula

a(n) = Sum_{k=1..n} k^n * A122843(n,k).

A230382 Number of ascending runs of length n in the permutations of [2n].

Original entry on oeis.org

1, 2, 21, 312, 5880, 133920, 3575880, 109549440, 3788104320, 145957593600, 6201593798400, 288084016281600, 14525808782284800, 790129980896256000, 46120599397192320000, 2875600728738017280000, 190740227037467627520000, 13411608375592258191360000
Offset: 0

Views

Author

Alois P. Heinz, Oct 17 2013

Keywords

Links

Crossrefs

A diagonal of A122843.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1+n,
      2*(2*n-1)*(n+1)*(n^2+n+1)*a(n-1)/((n+2)*(n^2-n+1)))
    end:
seq(a(n), n=0..25);

Formula

For n>0, a(n) = (n^2+n+1)*(2*n)!/((n+2)*n!). - Vaclav Kotesovec, Oct 18 2013

A141052 Number of runs or rising sequences of length 2 among all permutations of n.

Original entry on oeis.org

1, 4, 21, 130, 930, 7560, 68880, 695520, 7711200, 93139200, 1217462400, 17124307200, 257902444800, 4140968832000, 70614415872000, 1274546617344000, 24275666967552000, 486580401635328000, 10238462617743360000, 225651661258383360000, 5198503365971435520000
Offset: 2

Views

Author

Harlan J. Brothers, Jul 31 2008, Aug 24 2008

Keywords

Examples

			a[3]=4 because of the 6 permutations of n=3, there are 4 ascending runs of length 2:
{1,3} in {1,3,2}
{1,3} in {2,1,3}
{2,3} in {2,3,1}
{1,2} in {3,1,2}
a[3]=4 because of the 6 permutations of n=3, there are 4 rising sequences of length 2:
{1,2} in {1,3,2}
{2,3} in {2,1,3}
{2,3} in {2,3,1}
{1,2} in {3,1,2}

Links

Crossrefs

Column 2 of A122843.
Cf. A008292, A097900, A001286, A001048, A000142, A028387, A001710.

Programs

  • Mathematica
    Table[n!(5n + 1)/4! + Floor[2/n](1/12), {n, 2, 10}]

Formula

a(n) = n!*(5n+1)/4! + floor(2/n)*(1/12), n>=2.
Recurrence: a(n) = (n+1)*a(n-1)+(n-1)!/6, n>=2, with a(2)=1 and a(3)=4.
E.g.f.: x^2*(x-2)*(x-6)/(24*(x-1)^2).

Extensions

First example and typo in second example corrected by Harlan J. Brothers, Apr 29 2013
Previous Showing 11-16 of 16 results.

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