A182172 -id:A182172 - cp's OEIS frontend

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

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

Alois P. Heinz, Sep 30 2012

nonn

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

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

Alois P. Heinz, Table of n, a(n) for n = 5..1449 (terms 501..1000 from Seiichi Manyama)

Column k=5 of A047884.

Cf. A005817, A049401, A182172.

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);

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

Alois P. Heinz, Sep 30 2012

nonn

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

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

Alois P. Heinz, Table of n, a(n) for n = 6..400

Column k=6 of A047884.

Cf. A007579, A049401, A182172.

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

Alois P. Heinz, Sep 30 2012

nonn

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

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

Alois P. Heinz, Table of n, a(n) for n = 7..400

Column k=7 of A047884.

Cf. A007578, A007579, A182172.

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

Alois P. Heinz, Sep 30 2012

nonn

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

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

Alois P. Heinz, Table of n, a(n) for n = 8..400

Column k=8 of A047884.

Cf. A007578, A007580, A182172.

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

A218262 Number of standard Young tableaux of n cells and height >= 10.

Original entry on oeis.org

1, 11, 121, 1001, 8086, 59228, 426673, 2946593, 20161558, 135303408, 904408398, 5995379358, 39727129830, 262629161094, 1739604051411, 11535387587595, 76763703224070, 512448824337780, 3436760740882050, 23151339236295810, 156789753069685500, 1067435349046248600

Offset: 10

Alois P. Heinz, Oct 24 2012

nonn

Also number of self-inverse permutations in S_n with longest increasing subsequence of length >= 10.

Alois P. Heinz, Table of n, a(n) for n = 10..300
Wikipedia, Involution (mathematics)
Wikipedia, Young tableau

Column k=10 of A182222.

Cf. A000085, A212915, A182172.

a(n) = A000085(n) - A212915(n) = A182172(n,n) - A182172(n,9).

A218263 Number of standard Young tableaux of n cells and height >= 3.

Original entry on oeis.org

1, 4, 16, 56, 197, 694, 2494, 9244, 35234, 139228, 566788, 2387048, 10343101, 46193866, 211775002, 997265204, 4809609062, 23758479340, 119952340180, 618883933480, 3257842530546, 17492187873444, 95680438560276, 532985197799976, 3020676725917252

Offset: 3

Alois P. Heinz, Oct 24 2012

nonn

Also number of self-inverse permutations in S_n with longest increasing subsequence of length >= 3. a(3)=1: 123; a(4)=4: 1234, 1243, 1324, 2134.

Alois P. Heinz, Table of n, a(n) for n = 3..800
Wikipedia, Involution (mathematics)
Wikipedia, Young tableau

Column k=3 of A182222.

b:= proc(n) b(n):= `if`(n<2, 1, b(n-1) +(n-1)*b(n-2)) end:
a:= n-> b(n) -binomial(n, iquo(n, 2)):
seq(a(n), n=3..30);

b[n_] := b[n] = If[n<2, 1, b[n-1] + (n-1)*b[n-2]];
a[n_] := b[n] - Binomial[n, Quotient[n, 2]];
Table[a[n], {n, 3, 30}] (* Jean-François Alcover, Aug 23 2021, after Alois P. Heinz *)

a(n) = A000085(n) - A001405(n) = A182172(n,n) - A182172(n,2).

Conjecture: (n-6)*(n-3)*(n+1)*a(n) +(-n^3+6*n^2+11*n-36)*a(n-1) -(n-1)*(n^3-4*n^2-21*n+76)*a(n-2) +2*(n-1)*(n-2)*(3*n-19)*a(n-3) +4*(n-5)*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Jan 04 2017

A218264 Number of standard Young tableaux of n cells and height >= 4.

Original entry on oeis.org

1, 5, 25, 105, 441, 1785, 7308, 29898, 124641, 526669, 2276846, 10038964, 45353269, 209442533, 990777442, 4791502156, 23707812077, 119810145337, 618483875689, 3256714122209, 17488997849803, 95671400358075, 532959538382100, 3020603738202750, 17411069344112895

Offset: 4

Alois P. Heinz, Oct 24 2012

nonn

Also number of self-inverse permutations in S_n with longest increasing subsequence of length >= 4. a(4)=1: 1234; a(5)=5: 12345, 12354, 12435, 13245, 21345.

Alois P. Heinz, Table of n, a(n) for n = 4..800
Wikipedia, Involution (mathematics)
Wikipedia, Young tableau

Column k=4 of A182222.

a:= proc(n) option remember;
      `if`(n<5, [0$4, 1][n+1], ((-5-7*n+3*n^2)*a(n-1)
        +(n-1)*(n^2-n-11)*a(n-2) -2*n*(n-1)*(n-2)*a(n-3)
        -3*(n-1)*(n-2)*(n-3)*a(n-4))/((n+2)*(n-4)))
    end:
seq(a(n), n=4..30);

a[n_] := a[n] = If[n<5, {0,0,0,0,1}[[n+1]], ((-5-7n+3n^2)a[n-1] + (n-1)(n^2-n-11)a[n-2] - 2n(n-1)(n-2)a[n-3] - 3(n-1)(n-2)(n-3)a[n-4])/ ((n+2)(n-4))];
Table[a[n], {n, 4, 30}] (* Jean-François Alcover, Aug 23 2021, after Alois P. Heinz *)

a(n) = A000085(n) - A001006(n) = A182172(n,n) - A182172(n,3).

A218265 Number of standard Young tableaux of n cells and height >= 5.

Original entry on oeis.org

1, 6, 36, 176, 856, 3952, 18272, 83524, 384463, 1777010, 8304636, 39254076, 188160268, 915651672, 4527595824, 22771294440, 116496899100, 606656445480, 3214574890480, 17337658462800, 95128543350576, 530998366724576, 3013524116661952, 17385349086129304

Offset: 5

Alois P. Heinz, Oct 24 2012

nonn

Also number of self-inverse permutations in S_n with longest increasing subsequence of length >= 5. a(5)=1: 12345; a(6)=6: 123456, 123465, 123546, 124356, 132456, 213456.

Alois P. Heinz, Table of n, a(n) for n = 5..800
Wikipedia, Involution (mathematics)
Wikipedia, Young tableau

Column k=5 of A182222.

a:= proc(n) option remember; `if`(n<13,
      [0$5, 1, 6, 36, 176, 856, 3952, 18272, 83524][n+1],
      ((n^4-2*n^3-179*n^2+256*n+804) *a(n-1)
      +(n-1)*(n^4+6*n^3-295*n^2+1108*n+100) *a(n-2)
      -4*(n-1)*(n-2)*(6*n^2-83*n+67) *a(n-3)
      -16*(n-11)*(n-1)*(n-3)*(n-2)^2 *a(n-4))/
      ((n-12)*(n-5)*(n+4)*(n+3)))
    end:
seq(a(n), n=5..30);

a(n) = A000085(n) - A005817(n) = A182172(n,n) - A182172(n,4).

A218266 Number of standard Young tableaux of n cells and height >= 6.

Original entry on oeis.org

1, 7, 49, 273, 1506, 7788, 40161, 202917, 1028170, 5190328, 26375635, 134565795, 692890250, 3596739368, 18877483060, 100131220940, 537718999715, 2922918175965, 16100254700137, 89857257410905, 508473405642250, 2916903963927300, 16969580464205400

Offset: 6

Alois P. Heinz, Oct 24 2012

nonn

Also number of self-inverse permutations in S_n with longest increasing subsequence of length >= 6. a(6)=1: 123456; a(7)=7: 1234567, 1234576, 1234657, 1235467, 1243567, 1324567, 2134567.

Alois P. Heinz, Table of n, a(n) for n = 6..300
Wikipedia, Involution (mathematics)
Wikipedia, Young tableau

Column k=6 of A182222.

b:= proc(n) b(n):= `if`(n<2, 1, b(n-1) +(n-1)*b(n-2)) end:
g:= proc(n) option remember; `if`(n<3, [1, 1, 2][n+1],
      ((3*n^2+17*n+15)*g(n-1) +(n-1)*(13*n+9)*g(n-2)
       -15*(n-1)*(n-2)*g(n-3)) / ((n+4)*(n+6)))
    end:
a:= n-> b(n) -g(n):
seq(a(n), n=6..30);

a(n) = A000085(n) - A049401(n) = A182172(n,n) - A182172(n,5).

A218267 Number of standard Young tableaux of n cells and height >= 7.

Original entry on oeis.org

1, 8, 64, 400, 2465, 14092, 80016, 442248, 2442351, 13375366, 73477622, 403703404, 2230591660, 12380801756, 69225756076, 389806286920, 2213844625658, 12681996193252, 73339826141716, 428242854338216, 2526129602115517, 15056977593085444, 90712249806247400

Offset: 7

Alois P. Heinz, Oct 24 2012

nonn

Also number of self-inverse permutations in S_n with longest increasing subsequence of length >= 7. a(7)=1: 1234567; a(8)=8: 12345678, 12345687, 12345768, 12346578, 12354678, 12435678, 13245678, 21345678.

Alois P. Heinz, Table of n, a(n) for n = 7..300
Wikipedia, Involution (mathematics)
Wikipedia, Young tableau

Column k=7 of A182222.

b:= proc(n) b(n):= `if`(n<2, 1, b(n-1) +(n-1)*b(n-2)) end:
g:= proc(n) option remember;
      `if`(n<4, [1, 1, 2, 4][n+1], ((20*n^2+184*n+336)*g(n-1)
       +4*(n-1)*(10*n^2+58*n+33)*g(n-2) -144*(n-1)*(n-2)*g(n-3)
       -144*(n-1)*(n-2)*(n-3)*g(n-4)) / ((n+5)*(n+8)*(n+9)))
    end:
a:= n-> b(n) -g(n):
seq(a(n), n=7..30);

a(n) = A000085(n) - A007579(n) = A182172(n,n) - A182172(n,6).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A218262 Number of standard Young tableaux of n cells and height >= 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A218263 Number of standard Young tableaux of n cells and height >= 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A218264 Number of standard Young tableaux of n cells and height >= 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A218265 Number of standard Young tableaux of n cells and height >= 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A218266 Number of standard Young tableaux of n cells and height >= 6.