A350015 -id:A350015 - cp's OEIS frontend

A349979 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose second-longest cycle has length exactly k; n>=0, 0<=k<=floor(n/2).

1, 1, 1, 1, 2, 4, 6, 15, 3, 24, 61, 35, 120, 290, 270, 40, 720, 1646, 1974, 700, 5040, 11025, 14707, 8288, 1260, 40320, 85345, 117459, 90272, 29484, 362880, 749194, 1023390, 974720, 446040, 72576, 3628800, 7347374, 9813210, 10666480, 6332040, 2128896

Offset: 0

Steven Finch, Dec 07 2021

If the permutation has no second cycle, then its second-longest cycle is defined to have length 0.

			Triangle begins:
[0]     1;
[1]     1;
[2]     1,     1;
[3]     2,     4;
[4]     6,    15,      3;
[5]    24,    61,     35;
[6]   120,   290,    270,    40;
[7]   720,  1646,   1974,   700;
[8]  5040, 11025,  14707,  8288,  1260;
[9] 40320, 85345, 117459, 90272, 29484;
    ...

Alois P. Heinz, Rows n = 0..200, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Column 0 gives 1 together with A000142.
Column 1 gives 1 - (n-1)! + A006231(n).
Row sums give A000142.
T(2n,n) gives A110468(n-1) for n>=1.
Cf. A126074, A145877, A332851, A349980, A350015, A350016, A350273, A350274.

b:= proc(n, l) option remember; `if`(n=0, x^l[1], add((j-1)!*
      b(n-j, sort([l[], j])[2..3])*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n, [0$2])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 07 2021

b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[(j - 1)!*b[n - j, Sort[ Append[l, j]][[2 ;; 3]]]*Binomial[n - 1, j - 1], {j, 1, n}]];
T[n_] := With[{p = b[n, {0, 0}]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

Sum_{k=0..floor(n/2)} k * T(n,k) = A332851(n). - Alois P. Heinz, Dec 07 2021

A349980 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose second-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 6, 7, 3, 8, 24, 31, 15, 20, 30, 120, 191, 135, 40, 90, 144, 720, 1331, 945, 280, 420, 504, 840, 5040, 10655, 7077, 4480, 1260, 2688, 3360, 5760, 40320, 95887, 64197, 41552, 11340, 18144, 20160, 25920, 45360, 362880, 958879, 646965, 395360, 238140, 72576, 151200, 172800, 226800, 403200

Offset: 0

Steven Finch, Dec 07 2021

If the permutation has no second cycle, then its second-longest cycle is defined to have length 0.

			Triangle begins:
[0]     1;
[1]     1;
[2]     1,     1;
[3]     2,     1,     3;
[4]     6,     7,     3,     8;
[5]    24,    31,    15,    20,    30;
[6]   120,   191,   135,    40,    90,   144;
[7]   720,  1331,   945,   280,   420,   504,   840;
[8]  5040, 10655,  7077,  4480,  1260,  2688,  3360,  5760;
[9] 40320, 95887, 64197, 41552, 11340, 18144, 20160, 25920, 45360;
    ...

Alois P. Heinz, Rows n = 0..141, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Column 0 gives 1 together with A000142.
Column 1 gives the nonzero terms of A155521.
Row sums give A000142.
T(n,n-1) gives A059171(n) for n>=1.
Cf. A126074, A145877, A332906, A349979, A350015, A350016, A350273, A350274.

m:= infinity:
b:= proc(n, l) option remember; `if`(n=0, x^`if`(l[2]=m,
      0, l[2]), add(b(n-j, sort([l[], j])[1..2])
               *binomial(n-1, j-1)*(j-1)!, j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, n-1)))(b(n, [m$2])):
seq(T(n), n=0..10);  # Alois P. Heinz, Dec 07 2021

m = Infinity;
b[n_, l_] := b[n, l] = If[n == 0, x^If[l[[2]] == m, 0, l[[2]]], Sum[b[n-j, Sort[Append[l, j]][[1;;2]]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]];
T[n_] := With[{p = b[n, {m, m}]}, Table[Coefficient[p, x, i], {i, 0, Max[0, n - 1]}]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

Sum_{k=0..max(0,n-1)} k * T(n,k) = A332906(n). - Alois P. Heinz, Dec 07 2021

More terms from Alois P. Heinz, Dec 07 2021

A350016 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose third-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-2).

Original entry on oeis.org

1, 1, 2, 5, 1, 17, 1, 6, 74, 11, 15, 20, 394, 56, 60, 120, 90, 2484, 407, 525, 490, 630, 504, 18108, 3235, 4725, 2240, 4620, 4032, 3360, 149904, 29143, 40509, 27440, 26460, 33264, 30240, 25920, 1389456, 291394, 398790, 319760, 163800, 302400, 277200, 259200, 226800

Offset: 0

Steven Finch, Dec 08 2021

If the permutation has no third cycle, then its third-longest cycle is defined to have length 0.

			Triangle begins:
[0]      1;
[1]      1;
[2]      2;
[3]      5,     1;
[4]     17,     1,     6;
[5]     74,    11,    15,    20;
[6]    394,    56,    60,   120,    90;
[7]   2484,   407,   525,   490,   630,   504;
[8]  18108,  3235,  4725,  2240,  4620,  4032,  3360;
[9] 149904, 29143, 40509, 27440, 26460, 33264, 30240, 25920;
...

Alois P. Heinz, Rows n = 0..142, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Column 0 gives 1 together with A000774.
Column 1 gives the column 3 of A208956.
Row sums give A000142.
Cf. A126074, A145877, A332907, A349979, A349980, A350015, A350273, A350274.

m:= infinity:
b:= proc(n, l) option remember; `if`(n=0, x^`if`(l[3]=m,
      0, l[3]), add(b(n-j, sort([l[], j])[1..3])
               *binomial(n-1, j-1)*(j-1)!, j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [m$3])):
seq(T(n), n=0..10);  # Alois P. Heinz, Dec 11 2021

m = Infinity;
b[n_, l_] := b[n, l] = If[n == 0, x^If[l[[3]] == m, 0, l[[3]]], Sum[b[n-j, Sort[Append[l, j]][[1;;3]]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]];
T[n_] := With[{p = b[n, {m, m, m}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

Sum_{k=0..n-2} k * T(n,k) = A332907(n) for n >= 3. - Alois P. Heinz, Dec 12 2021

A350273 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose fourth-longest cycle has length exactly k; n >= 0, 0 <= k <= floor(n/4).

Original entry on oeis.org

1, 1, 2, 6, 23, 1, 109, 11, 619, 101, 4108, 932, 31240, 8975, 105, 268028, 91387, 3465, 2562156, 991674, 74970, 27011016, 11514394, 1391390, 311378616, 143188574, 24188010, 246400, 3897004032, 1905067958, 412136010, 12812800, 52626496896, 27059601596, 7053834788, 438357920

Offset: 0

Steven Finch, Dec 22 2021

If the permutation has no fourth cycle, then its fourth-longest cycle is defined to have length 0.

			Triangle begins:
[0]      1;
[1]      1;
[2]      2;
[3]      6;
[4]     23,     1;
[5]    109,    11;
[6]    619,   101;
[7]   4108,   932;
[8]  31240,  8975,  105;
[9] 268028, 91387, 3465;
    ...

Alois P. Heinz, Rows n = 0..100, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Row sums give A000142(n).

Cf. A126074, A145877, A332853, A349979, A349980, A350015, A350016, A350274.

b:= proc(n, l) option remember; `if`(n=0, x^l[1], add((j-1)!*
      b(n-j, sort([l[], j])[2..5])*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$4])):
seq(T(n), n=0..14);  # Alois P. Heinz, Dec 22 2021

b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[(j - 1)!*b[n - j, Sort[ Append[l, j]][[2 ;; 5]]]*Binomial[n - 1, j - 1], {j, 1, n}]];
T[n_] := With[{p = b[n, {0, 0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 29 2021, after Alois P. Heinz *)

Sum_{k=0..floor(n/4)} k * T(n,k) = A332853(n) for n >= 4.

More terms from Alois P. Heinz, Dec 22 2021

A350274 Triangle read by rows: T(n,k) is the number of n-permutations whose fourth-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-3).

Original entry on oeis.org

1, 1, 2, 6, 23, 1, 109, 1, 10, 619, 16, 45, 40, 4108, 92, 210, 420, 210, 31240, 771, 1645, 2800, 2520, 1344, 268028, 6883, 17325, 15960, 26460, 18144, 10080, 2562156, 68914, 173250, 148400, 226800, 211680, 151200, 86400, 27011016, 757934, 1854930, 1798720, 1801800, 2494800, 1940400, 1425600, 831600

Offset: 0

Steven Finch, Dec 22 2021

If the permutation has no fourth cycle, then its fourth-longest cycle is defined to have length 0.

			Triangle begins:
[0]      1;
[1]      1;
[2]      2;
[3]      6;
[4]     23,    1;
[5]    109,    1,    10;
[6]    619,   16,    45,    40;
[7]   4108,   92,   210,   420,   210;
[8]  31240,  771,  1645,  2800,  2520,  1344;
[9] 268028, 6883, 17325, 15960, 26460, 18144, 10080;
    ...

Alois P. Heinz, Rows n = 0..100, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Column 0 is 1 for n=0, together with A000142(n) - A122105(n-1) for n>=1.
Row sums give A000142.
Cf. A126074, A145877, A332908, A349979, A349980, A350015, A350016, A350273.

m:= infinity:
b:= proc(n, l) option remember; `if`(n=0, x^`if`(l[4]=m,
      0, l[4]), add(b(n-j, sort([l[], j])[1..4])
               *binomial(n-1, j-1)*(j-1)!, j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [m$4])):
seq(T(n), n=0..11);  # Alois P. Heinz, Dec 22 2021

m = Infinity;
b[n_, l_] := b[n, l] = If[n == 0, x^If[l[[4]] == m, 0, l[[4]]], Sum[b[n-j, Sort[Append[l, j]][[1 ;; 4]]]*Binomial[n-1, j-1]*(j-1)!, {j, 1, n}]];
T[n_] := With[{p = b[n, {m, m, m, m}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Dec 29 2021, after Alois P. Heinz *)

Sum_{k=0..n-3} k * T(n,k) = A332908(n) for n >= 4.

More terms from Alois P. Heinz, Dec 22 2021

A332852 Number of entries in the third cycles of all permutations of [n] when cycles are ordered by decreasing lengths.

Original entry on oeis.org

1, 7, 46, 341, 2871, 26797, 275353, 3090544, 37652660, 495756306, 7015094802, 106125820737, 1710625964061, 29267936828691, 529655709670675, 10110999740354242, 203072647138681534, 4280118000323963708, 94470690960204259548, 2179212745888578818307

Offset: 3

Alois P. Heinz, Feb 26 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..450
Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
Wikipedia, Permutation

Column k=3 of A322384.

Cf. A350015.

b:= proc(n, l) option remember; `if`(n=0, l[3], add((j-1)!*b(n-j,
      sort([l[], j], `>`)[1..3])*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, [0$3]):
seq(a(n), n=3..23);

b[n_, l_] := b[n, l] = If[n == 0, l[[3]], Sum[(j-1)! b[n-j, ReverseSort[ Append[l, j]][[1 ;; 3]]] Binomial[n - 1, j - 1], {j, 1, n}]];
a[n_] := b[n, {0, 0, 0}];
a /@ Range[3, 23] (* Jean-François Alcover, Mar 01 2020, after Alois P. Heinz *)

a(n) = Sum_{k=0..floor(n/3)} k * A350015(n,k). - Alois P. Heinz, Dec 12 2021

cp's OEIS Frontend

A349979 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose second-longest cycle has length exactly k; n>=0, 0<=k<=floor(n/2).

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A349980 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose second-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-1).

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A350016 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose third-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-2).

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A350273 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose fourth-longest cycle has length exactly k; n >= 0, 0 <= k <= floor(n/4).

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A350274 Triangle read by rows: T(n,k) is the number of n-permutations whose fourth-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-3).

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A332852 Number of entries in the third cycles of all permutations of [n] when cycles are ordered by decreasing lengths.

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