A350275 -id:A350275 - cp's OEIS frontend

A350078 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose second-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/2).

1, 1, 3, 1, 17, 10, 142, 87, 27, 1569, 911, 645, 21576, 11930, 10260, 2890, 355081, 189610, 174132, 104720, 6805296, 3543617, 3229275, 2493288, 705740, 148869153, 76060087, 67843521, 60223520, 34424208, 3660215680, 1842497914, 1605373560, 1530575960, 1051155000, 310181886

Offset: 0

Steven Finch, Dec 12 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no second component, then its second-largest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       3,      1;
      17,     10;
     142,     87,     27;
    1569,    911,    645;
   21576,  11930,  10260,   2890;
  355081, 189610, 174132, 104720;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Column 0 gives gives 1 together with A001865.
Row sums give A000312.
Cf. A001865, A350079, A350080, A350081, A350275, A350276.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^l[1], add(g(i)*
      b(n-i, sort([l[], i])[-2..-1])*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$2])):
seq(T(n), n=0..10);  # Alois P. Heinz, Dec 17 2021

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := g[n, l] = If[n == 0, x^l[[1]], Sum[g[i]*b[n - i, Sort[ Append[l, i]][[-2 ;; -1]]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {0, 0}]}, 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 *)

More terms (three rows) from Alois P. Heinz, Dec 15 2021

A350079 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose second-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-1).

Original entry on oeis.org

1, 1, 3, 1, 17, 1, 9, 142, 19, 27, 68, 1569, 201, 135, 510, 710, 21576, 2921, 3465, 2890, 6390, 9414, 355081, 50233, 63630, 20230, 84490, 98847, 151032, 6805296, 1004599, 1196181, 918680, 705740, 1493688, 1812384, 2840648, 148869153, 22872097, 26904339, 23943752, 6351660, 28072548, 30810528, 38348748, 61247664

Offset: 0

Steven Finch, Dec 12 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no second component, then its second-smallest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       3,     1;
      17,     1,     9;
     142,    19,    27,    68;
    1569,   201,   135,   510,   710;
   21576,  2921,  3465,  2890,  6390,  9414;
  355081, 50233, 63630, 20230, 84490, 98847, 151032;
  ...

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 gives 1 together with A001865.
Row sums give A000312.
Cf. A001865, A350078, A350080, A350081, A350275, A350276

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[2],
      add(b(n-i, sort([l[], i])[1..2])*g(i)*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$2])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 17 2021

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

More terms (two rows) from Alois P. Heinz, Dec 15 2021

A350080 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/3).

Original entry on oeis.org

1, 1, 4, 26, 1, 237, 19, 2789, 336, 40270, 5981, 405, 689450, 115193, 18900, 13657756, 2459955, 659505, 307348641, 58366045, 20330163, 1375640, 7745565616, 1530739594, 623758590, 99936200, 216114310994, 44076571672, 19795671225, 5325116720

Offset: 0

Steven Finch, Dec 12 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no third component, then its third-largest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       4;
      26,     1;
     237,    19;
    2789,   336;
   40270,   5981,   405;
  689450, 115193, 18900;
  ...

Alois P. Heinz, Rows n = 0..120, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Row sums give A000312.

Cf. A001865, A350078, A350079, A350081, A350275, A350276.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^l[1], add(g(i)*
      b(n-i, sort([l[], i])[-3..-1])*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$3])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 17 2021

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

More terms (4 rows) from Alois P. Heinz, Dec 16 2021

A350081 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-2).

Original entry on oeis.org

1, 1, 4, 26, 1, 237, 1, 18, 2789, 31, 135, 170, 40270, 386, 810, 3060, 2130, 689450, 6574, 13545, 36295, 44730, 32949, 13657756, 129291, 327285, 323680, 944300, 790776, 604128, 307348641, 2910709, 7207137, 6602120, 15476580, 18780930, 16311456, 12782916

Offset: 0

Steven Finch, Dec 12 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no third component, then its third-smallest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       4;
      26,    1;
     237,    1,    18;
    2789,   31,   135,   170;
   40270,  386,   810,  3060,  2130;
  689450, 6574, 13545, 36295, 44730, 32949;
  ...

Alois P. Heinz, Rows n = 0..120, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Row sums give A000312.

Cf. A001865, A350078, A350079, A350080, A350275, A350276.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[3],
      add(b(n-i, sort([l[], i])[1..3])*g(i)*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$3])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 17 2021

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

More terms (two rows) from Alois P. Heinz, Dec 16 2021

A350276 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-3).

Original entry on oeis.org

1, 1, 4, 27, 255, 1, 3094, 1, 30, 45865, 46, 405, 340, 803424, 659, 3780, 10710, 4970, 16239720, 12867, 48405, 209440, 178920, 87864, 372076163, 284785, 1225665, 3005940, 5457060, 3558492, 1812384, 9529560676, 7126384, 32262300, 51205700, 135084600, 120593340, 81557280, 42609720

Offset: 0

Steven Finch, Dec 22 2021

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no fourth component, then its fourth-smallest component is defined to have size 0.

			Triangle begins:
       1;
       1;
       4;
      27;
     255,   1;
    3094,   1,   30;
   45865,  46,  405,   340;
  803424, 659, 3780, 10710, 4970;
  ...

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 A000312.

Cf. A001865, A350078, A350079, A350080, A350081, A350275.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[4],
      add(b(n-i, sort([l[], i])[1..4])*g(i)*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$4])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 22 2021

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^(l /. Infinity -> 0)[[4]], Sum[b[n - i, Sort[Append[l, i]][[1 ;; 4]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, Table[Infinity, {4}]]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

cp's OEIS Frontend

A350078 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose second-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/2).

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A350079 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose second-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-1).

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A350080 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/3).

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Maple

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A350081 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-2).

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Examples

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Crossrefs

Programs

Maple

Mathematica

Extensions

A350276 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica