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.

Showing 1-5 of 5 results.

A008304 Triangle read by rows: T(n,k) (n>=1; 1<=k<=n) is the number of permutations of [n] in which the longest increasing run has length k.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 16, 6, 1, 1, 69, 41, 8, 1, 1, 348, 293, 67, 10, 1, 1, 2016, 2309, 602, 99, 12, 1, 1, 13357, 19975, 5811, 1024, 137, 14, 1, 1, 99376, 189524, 60875, 11304, 1602, 181, 16, 1, 1, 822040, 1960041, 690729, 133669, 19710, 2360, 231, 18, 1, 1, 7477161
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Row n has n terms.

Examples

			Triangle T(n,k) begins:
  1;
  1,   1;
  1,   4,   1;
  1,  16,   6,  1;
  1,  69,  41,  8,  1;
  1, 348, 293, 67, 10,  1;
  ...
T(3,2) = 4 because we have (13)2, 2(13), (23)1, 3(12), where the parentheses surround runs of length 2.

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261, Table 7.4.1.

Links

Crossrefs

Row sums give A000142. Sum_{k=1..n} k*T(n,k) = A064314(n). Cf. A064315.
Columns k=1-10 give: A000012, A000303, A000402, A000434, A000456, A000467, A230055, A230234, A230235, A230236.
T(2n+j,n+j) for j=0-10 gives: A230341, A230251, A230342, A230343, A230344, A230345, A230346, A230347, A230348, A230349, A230350.

Programs

  • Maple
    b:= proc(u, o, t, k) option remember; `if`(t=k, (u+o)!,
      `if`(max(t, u)+o b(0, n, 0, k) -b(0, n, 0, k+1):
seq(seq(T(n,k), k=1..n), n=1..15);  # Alois P. Heinz, Oct 16 2013
  • Mathematica
    b[u_, o_, t_, k_] := b[u, o, t, k] = If[t == k, (u + o)!, If[Max[t, u]+o < k, 0, Sum[b[u+j-1, o-j, t+1, k], {j, 1, o}] + Sum[b[u-j, o+j-1, 1, k], {j, 1, u}]]]; T[n_, k_] := b[0, n, 0, k] - b[0, n, 0, k+1]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 10 2014, translated from Alois P. Heinz's Maple code *)
(*additional code*)
nn=12;a[r_]:=Apply[Plus,Table[Normal[Series[y x^(r+1)/(1-Sum[y x^i,{i,1,r}]),{x,0,nn}]][[n]]/(n+r)!,{n,1,nn-r}]]/.y->-1;Map[Select[#,#>0&]&,Transpose[Prepend[Table[Drop[Range[0,nn]! CoefficientList[Series[1/(1-x-a[n+1])-1/(1-x-a[n]),{x,0,nn}],x],1],{n,1,8}],Table[1,{nn}]]]]//Grid (* Geoffrey Critzer, Feb 25 2014 *)

Formula

E.g.f. of column k: 1/Sum_{n>=0} ((k+1)*n+1-x)*x^((k+1)*n)/((k+1)*n+1)! - 1/Sum_{n>=0} (k*n+1-x)*x^(k*n)/(k*n+1)!. - Alois P. Heinz, Oct 13 2013
T(n,k) = A122843(n,k) for k > n/2. - Alois P. Heinz, Oct 17 2013

Extensions

More terms from David W. Wilson, Sep 07 2001
Better description from Emeric Deutsch, May 08 2004

A000303 Number of permutations of [n] in which the longest increasing run has length 2.

Original entry on oeis.org

0, 1, 4, 16, 69, 348, 2016, 13357, 99376, 822040, 7477161, 74207208, 797771520, 9236662345, 114579019468, 1516103040832, 21314681315997, 317288088082404, 4985505271920096, 82459612672301845, 1432064398910663704, 26054771465540507272
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			a(3)=4 because we have (13)2, 2(13), (23)1, 3(12), where the parentheses surround increasing runs of length 2.

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261, Table 7.4.1.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column 2 of A008304. Other columns: A000402, A000434, A000456, A000467, A230055.
Cf. A001250, A001251, A001252, A001253, A010026, A211318.
Equals 1 less than A049774. - Greg Dresden, Feb 22 2020

Programs

  • Mathematica
    b[u_, o_, t_, k_] := b[u, o, t, k] = If[t == k, (u + o)!, If[Max[t, u] + o < k, 0, Sum[b[u + j - 1, o - j, t + 1, k], {j, 1, o}] + Sum[b[u - j, o + j - 1, 1, k], {j, 1, u}]]];
T[n_, k_] := b[0, n, 0, k] - b[0, n, 0, k + 1];
a[n_] := T[n, 2];
Array[a, 30] (* Jean-François Alcover, Jul 19 2018, after Alois P. Heinz *)

Extensions

Better description from Emeric Deutsch, May 08 2004
Edited and extended by Max Alekseyev, May 20 2012

A000402 Number of permutations of [n] in which the longest increasing run has length 3.

Original entry on oeis.org

0, 0, 1, 6, 41, 293, 2309, 19975, 189524, 1960041, 21993884, 266361634, 3465832370, 48245601976, 715756932697, 11277786883720, 188135296651083, 3313338641692957, 61444453534759589, 1196988740015236617, 24442368179977776766, 522124104504306695929
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			a(4)=6 because we have (124)3, (134)2, (234)1, 4(123), 3(124) and 2(134), where the parentheses surround increasing runs of length 3.

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261, Table 7.4.1. (Values for n>=16 are incorrect.)
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column 3 of A008304. Other columns: A000303, A000434, A000456, A000467.
Cf. A001250, A001251, A001252, A001253, A010026, A211318.

Programs

  • Mathematica
    b[u_, o_, t_, k_] := b[u, o, t, k] = If[t == k, (u + o)!, If[Max[t, u] + o < k, 0, Sum[b[u + j - 1, o - j, t + 1, k], {j, 1, o}] + Sum[b[u - j, o + j - 1, 1, k], {j, 1, u}]]];
T[n_, k_] := b[0, n, 0, k] - b[0, n, 0, k + 1];
a[n_] := T[n, 3];
Array[a, 30] (* Jean-François Alcover, Jul 19 2018, after Alois P. Heinz *)

Extensions

Better description from Emeric Deutsch, May 08 2004
Terms a(16), a(17) are corrected and further terms added by Max Alekseyev, May 20 2012

A000456 Number of permutations of [n] in which the longest increasing run has length 5.

Original entry on oeis.org

0, 0, 0, 0, 1, 10, 99, 1024, 11304, 133669, 1695429, 23023811, 333840443, 5153118154, 84426592621, 1463941342191, 26793750988542, 516319125748337, 10451197169218523, 221738082618710329, 4921234092461339819, 114041894068935641488
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			a(6)=10 because we have (12346)5, (12356)4, (12456)3, (13456)2, (23456)1, 6(12345), 5(12346), 4(12356), 3(12456) and 2(13456), where the parentheses surround increasing runs of length 5.

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column 5 of A008304. Other columns: A000303, A000402, A000434, A000467.
Cf. A001250, A001251, A001252, A001253, A010026, A211318.

Programs

  • Mathematica
    b[u_, o_, t_, k_] := b[u, o, t, k] = If[t == k, (u + o)!, If[Max[t, u] + o < k, 0, Sum[b[u + j - 1, o - j, t + 1, k], {j, 1, o}] + Sum[b[u - j, o + j - 1, 1, k], {j, 1, u}]]]; T[n_, k_] := b[0, n, 0, k] - b[0, n, 0, k + 1]; a[n_] := T[n, 5]; Array[a, 25] (* Jean-François Alcover, Feb 08 2016, after Alois P. Heinz in A008304 *)

Extensions

Better description from Emeric Deutsch, May 08 2004
Edited and extended by Max Alekseyev, May 20 2012

A000467 Number of permutations of [n] in which the longest increasing run has length 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 12, 137, 1602, 19710, 257400, 3574957, 52785901, 827242933, 13730434111, 240806565782, 4452251786946, 86585391630673, 1767406549387381, 37790452850585180, 844817788372455779, 19711244788916894489, 479203883157602851294
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column 6 of A008304. Other columns: A000303, A000402, A000434, A000456.
Cf. A001250, A001251, A001252, A001253, A010026, A211318.

Programs

  • Mathematica
    b[u_, o_, t_, k_] := b[u, o, t, k] = If[t == k, (u + o)!, If[Max[t, u] + o < k, 0, Sum[b[u + j - 1, o - j, t + 1, k], {j, 1, o}] + Sum[b[u - j, o + j - 1, 1, k], {j, 1, u}]]]; T[n_, k_] := b[0, n, 0, k] - b[0, n, 0, k + 1]; a[n_] := T[n, 6]; Array[a, 23] (* Jean-François Alcover, Feb 08 2016, after Alois P. Heinz in A008304 *)

Extensions

Edited and extended by Max Alekseyev, May 20 2012
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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