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.

Previous Showing 11-18 of 18 results.

A217321 Number of self-inverse permutations in S_n with longest increasing subsequence of length 9.

Original entry on oeis.org

1, 9, 89, 639, 4655, 30330, 198148, 1233743, 7694099, 46938514, 287070944, 1738940782, 10570927022, 64059763010, 389873574058, 2373799261605, 14522526860316, 89060416668016, 548932942208392, 3395326330414774, 21109553761623110, 131785019270029876
Offset: 9

Views

Author

Alois P. Heinz, Sep 30 2012

Keywords

Comments

Also the number of Young tableaux with n cells and 9 rows.

Links

Crossrefs

Column k=9 of A047884.
Cf. A007580, A182172, A212915.

Formula

a(n) = A182172(n,9)-A182172(n,8) = A212915(n)-A007580(n).

A217322 Number of self-inverse permutations in S_n with longest increasing subsequence of length 10.

Original entry on oeis.org

1, 10, 109, 857, 6798, 48007, 338529, 2267425, 15164662, 98964444, 645978814, 4168541022, 26949303558, 173445855265, 1119737108943, 7224864497439, 46800745943134, 303692912870933, 1979556048016406, 12943419575576650, 85040314513698164, 560910092712436079
Offset: 10

Views

Author

Alois P. Heinz, Sep 30 2012

Keywords

Comments

Also the number of Young tableaux with n cells and 10 rows.

Links

Crossrefs

Column k=10 of A047884.
Cf. A182172, A212915, A212916.

Formula

a(n) = A182172(n,10)-A182172(n,9) = A212916(n)-A212915(n).

A217324 Number of self-inverse permutations in S_n with longest increasing subsequence of length 4.

Original entry on oeis.org

1, 4, 19, 69, 265, 929, 3356, 11626, 41117, 142206, 499836, 1734328, 6099193, 21282265, 75125770, 263906332, 936517637, 3313246237, 11827430209, 42139231729, 151339387003, 542857007499, 1961171657524, 7079621540798, 25720257983591, 93396276789196
Offset: 4

Views

Author

Alois P. Heinz, Sep 30 2012

Keywords

Comments

Also the number of Young tableaux with n cells and 4 rows.

Examples

			a(4) = 1: 1234.
a(5) = 4: 12354, 12435, 13245, 21345.
a(6) = 19: 123654, 124365, 125436, 125634, 126453, 132465, 132546, 143256, 145236, 153426, 163452, 213465, 213546, 214356, 321456, 341256, 423156, 523416, 623451.

Links

Crossrefs

Column k=4 of A047884.
Cf. A001006, A005817, A182172.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<4, 0, `if`(n=4, 1,
      ((2+n)*(30*n^5+199*n^4-374*n^3-1537*n^2-406*n+408)*a(n-1)
       -4*(n-1)*(n-2)*(120*n^4+46*n^3-471*n^2+371*n+204)*a(n-3)
       +(n-1)*(285*n^5-262*n^4-2755*n^3-1520*n^2+820*n-48)*a(n-2)
       -48*(n-1)*(n-3)*(3*n+7)*(5*n+4)*(n-2)^2*a(n-4))/
      ((n-4)*(5*n-1)*(3*n+4)*(n+4)*(n+3)*(n+2))))
    end:
seq(a(n), n=4..40);
  • Mathematica
    h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := g[n, i, l] = If[n == 0 || i == 1, Function[p, h[p]*x^If[p == {}, 0, p[[1]]]][Join[l, Array[1&, n]]], Sum[g[n - i*j, i - 1, Join[l, Array[i&, j]]], {j, 0, n/i}]];
a[n_] := a[n] = Coefficient[g[n, n, {}], x, 4];
Table[Print[n, " ", a[n]]; a[n], {n, 4, 40}]
(* or: *)
MotzkinNumber = DifferenceRoot[Function[{y, n}, {(-3n-3)*y[n] + (-2n-5)*y[n+1] + (n+4)*y[n+2] == 0, y[0] == 1, y[1] == 1}]];
a[n_] := CatalanNumber[Quotient[n+1, 2]]*CatalanNumber[Quotient[n+2, 2]] - MotzkinNumber[n];
Table[a[n], {n, 4, 40}]
(* Jean-François Alcover, Oct 27 2021, after Alois P. Heinz in A047884 and second formula *)

Formula

a(n) = A182172(n,4)-A182172(n,3) = A005817(n)-A001006(n).

A217325 Number of self-inverse permutations in S_n with longest increasing subsequence of length 5.

Original entry on oeis.org

1, 5, 29, 127, 583, 2446, 10484, 43363, 181546, 748840, 3114308, 12878441, 53594473, 222761422, 930856456, 3893811380, 16365678160, 68937445765, 291656714515, 1237403762663, 5271285939671, 22524961082326, 96620152734652, 415768621923904, 1795530067804295
Offset: 5

Views

Author

Alois P. Heinz, Sep 30 2012

Keywords

Comments

Also the number of Young tableaux with n cells and 5 rows.

Examples

			a(5) = 1: 12345.
a(6) = 5: 123465, 123546, 124356, 132456, 213456.

Links

Crossrefs

Column k=5 of A047884.
Cf. A005817, A049401, A182172.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<5, 0, `if`(n=5, 1,
     ((n+3)*(166075637*n^5+3319452867*n^4+10706068615*n^3-39910302747*n^2
       -182846631872*n-159926209260)*a(n-1) +(840221898216*n+133982123900
       -322021480097*n^3-83890810854*n^4+12016871251*n^5+3735622433*n^6
       +111397917411*n^2)*a(n-2)-(n-2)*(2142183361*n^5+66617759078*n^4
       -47640468971*n^3-611402096064*n^2+15449945364*n+452645243780)*a(n-3)
       -(n-2)*(n-3)*(33769818805*n^4-54918997862*n^3 -469629276839*n^2
       +789889969148*n +94438295920)*a(-4+n) -4*(n-2)*(n-3)*(-4+n)*
       (2060107324*n^3 -87569131518*n^2+293565842963*n -151080184425)*a(n-5)
       +240*(n-2)*(n-3)*(n-5)*(168175627*n-312397451)*(-4+n)^2*a(n-6))/
       (8*(13927136*n+37088781)*(n-5)*(n+6)*(n+4)*(n+3)^2)))
    end:
seq(a(n), n=5..40);

Formula

a(n) = A182172(n,5) - A182172(n,4) = A049401(n) - A005817(n).

A217326 Number of self-inverse permutations in S_n with longest increasing subsequence of length 6.

Original entry on oeis.org

1, 6, 41, 209, 1106, 5323, 26069, 122901, 585922, 2747977, 13000269, 61088173, 289186846, 1366147708, 6496681304, 30905464864, 147912712795, 709073550307, 3418258506885, 16517431269189, 80230551304034, 390774361811783, 1912602871119956, 9388456361080840
Offset: 6

Views

Author

Alois P. Heinz, Sep 30 2012

Keywords

Comments

Also the number of Young tableaux with n cells and 6 rows.

Examples

			a(6) = 1: 123456.
a(7) = 6: 1234576, 1234657, 1235467, 1243567, 1324567, 2134567.

Links

Crossrefs

Column k=6 of A047884.
Cf. A007579, A049401, A182172.

Formula

a(n) = A182172(n,6)-A182172(n,5) = A007579(n)-A049401(n).

A217327 Number of self-inverse permutations in S_n with longest increasing subsequence of length 7.

Original entry on oeis.org

1, 7, 55, 319, 1904, 10275, 56135, 294386, 1556323, 8086433, 42298721, 219795160, 1149139210, 5999688692, 31506046052, 165664633982, 875886376212, 4643488263933, 24746018418741, 132328997879066, 711142850556217, 3836134976520394, 20791024498584110
Offset: 7

Views

Author

Alois P. Heinz, Sep 30 2012

Keywords

Comments

Also the number of Young tableaux with n cells and 7 rows.

Examples

			a(7) = 1: 1234567.
a(8) = 7: 12345687, 12345768, 12346578, 12354678, 12435678, 13245678, 21345678.

Links

Crossrefs

Column k=7 of A047884.
Cf. A007578, A007579, A182172.

Formula

a(n) = A182172(n,7)-A182172(n,6) = A007578(n)-A007579(n).

A217328 Number of self-inverse permutations in S_n with longest increasing subsequence of length 8.

Original entry on oeis.org

1, 8, 71, 461, 3057, 18225, 109446, 628652, 3628517, 20538209, 116808172, 659078098, 3737763884, 21153403644, 120354760098, 685455514294, 3925104616303, 22535893275064, 130089736567064, 753604985013128, 4388755545268226, 25660332309744370, 150802834643569274
Offset: 8

Views

Author

Alois P. Heinz, Sep 30 2012

Keywords

Comments

Also the number of Young tableaux with n cells and 8 rows.

Examples

			a(8) = 1: 12345678.
a(9) = 8: 123456798, 123456879, 123457689, 123465789, 123546789, 124356789, 132456789, 213456789.

Links

Crossrefs

Column k=8 of A047884.
Cf. A007578, A007580, A182172.

Formula

a(n) = A182172(n,8)-A182172(n,7) = A007580(n)-A007578(n).

A178249 Table T(n,k) counts the involutions of n with longest increasing contiguous subsequence of length k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 2, 1, 1, 14, 8, 2, 1, 1, 37, 27, 8, 2, 1, 1, 96, 94, 30, 8, 2, 1, 1, 270, 338, 114, 30, 8, 2, 1, 1, 777, 1237, 446, 118, 30, 8, 2, 1, 1, 2370, 4684, 1809, 473, 118, 30, 8, 2, 1, 1, 7450, 18142, 7502, 1964, 478, 118, 30, 8, 2, 1, 1, 24485, 72524, 32093, 8414, 1998, 478, 118, 30, 8, 2, 1
Offset: 1

Views

Author

Wouter Meeussen, Dec 20 2010

Keywords

Comments

Reverse of rows converges to 1,2,8,30,118,478,2004,8666,..

Examples

			T(4,2) = 6 because the 6 involutions with longest increasing contiguous subsequence lengths equal to 2 are: 1324, 1432, 2143, 3214, 3412, 4231.
Table starts:
1;
1,   1;
1,   2,   1;
1,   6,   2,   1;
1,  14,   8,   2,  1;
1,  37,  27,   8,  2, 1;
1,  96,  94,  30,  8, 2, 1;
1, 270, 338, 114, 30, 8, 2, 1;

Crossrefs

Cf. A008304; row sums are A000085; A047884 removes the contiguity requirement.

Programs

  • Mathematica
    (* first do *)
Needs["Combinatorica`"]
(* then *)
maxISS[perm_List] := Max[0, (Max @@ (Length[#1]*Sign[First[#1]] & ) /@ Split[Sign[Rest[#1] - Drop[#1, -1]]] & )[perm]];classMaxISS[par_?PartitionQ]:=Count[ maxISS/@( TableauxToPermutation[FirstLexicographicTableau[par], #]&/@Tableaux[par]  ) ,#]&/@(-1+Range[ Tr[par] ]);
Table[Apply[Plus,classMaxISS/@Partitions[n]],{n,2,6}];

Extensions

Definition corrected by Wouter Meeussen, Dec 22 2010
Previous Showing 11-18 of 18 results.

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