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.

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A047884 Triangle of numbers a(n,k) = number of Young tableaux with n cells and k rows (1 <= k <= n); also number of self-inverse permutations on n letters in which the length of the longest scattered (i.e., not necessarily contiguous) increasing subsequence is k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 9, 11, 4, 1, 1, 19, 31, 19, 5, 1, 1, 34, 92, 69, 29, 6, 1, 1, 69, 253, 265, 127, 41, 7, 1, 1, 125, 709, 929, 583, 209, 55, 8, 1, 1, 251, 1936, 3356, 2446, 1106, 319, 71, 9, 1, 1, 461, 5336, 11626, 10484, 5323, 1904, 461, 89, 10, 1
Offset: 1

Views

Author

Wouter Meeussen

Keywords

Examples

			For n=3 the 4 tableaux are
  1 2 3 . 1 2 . 1 3 . 1
  . . . . 3 . . 2 . . 2
  . . . . . . . . . . 3
Triangle begins:
  1;
  1,   1;
  1,   2,    1;
  1,   5,    3,     1;
  1,   9,   11,     4,     1;
  1,  19,   31,    19,     5,    1;
  1,  34,   92,    69,    29,    6,    1;
  1,  69,  253,   265,   127,   41,    7,   1;
  1, 125,  709,   929,   583,  209,   55,   8,  1;
  1, 251, 1936,  3356,  2446, 1106,  319,  71,  9,  1;
  1, 461, 5336, 11626, 10484, 5323, 1904, 461, 89, 10,  1;
  ...

References

  • W. Fulton, Young Tableaux, Cambridge, 1997.
  • D. Stanton and D. White, Constructive Combinatorics, Springer, 1986.

Links

Crossrefs

Row sums give A000085.
Cf. A049400, A049401, and A178249 which imposes contiguity.
Columns k=1-10 give: A000012, A014495, A217323, A217324, A217325, A217326, A217327, A217328, A217321, A217322. - Alois P. Heinz, Oct 03 2012
a(2n,n) gives A267436.

Programs

  • Maple
    h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) `if`(n=0 or i=1, (p->h(p)*x^`if`(p=[], 0, p[1]))
      ([l[], 1$n]), add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(g(n$2, [])):
seq(T(n), n=1..14); # Alois P. Heinz, Apr 16 2012, revised Mar 05 2014
  • Mathematica
    Table[ Plus@@( NumberOfTableaux/@ Reverse/@Union[ Sort/@(Compositions[ n-m, m ]+1) ]), {n, 12}, {m, n} ]
(* Second program: *)
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_] := 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}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][g[n, n, {}]];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

Extensions

Definition amended ('scattered' added) by Wouter Meeussen, Dec 22 2010

A267433 Number of permutations of [2n] with longest increasing subsequence of length n.

Original entry on oeis.org

1, 1, 13, 381, 17557, 1100902, 87116283, 8312317976, 927716186325, 118504614869214, 17044414451764396, 2725298085020712539, 479491040778079234419, 92050364310704637832186, 19146538134094625864605786, 4289203871330156652985437480
Offset: 0

Views

Author

Alois P. Heinz, Jan 15 2016

Keywords

Examples

			a(2) = 13: 1432, 2143, 2413, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312.

Links

Crossrefs

Cf. A047874, A126065, A230341, A267436.

Programs

  • Maple
    h:= proc(l) local n; n:= nops(l); add(i, i=l)! /mul(mul(1+l[i]-j+
    add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
                add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
a:= n-> g(n$2, [n]):
seq(a(n), n=0..20);
  • 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, h[Join[l, Table[1, {n}]]]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Table[i, {j}]]], {j, 0, n/i}]]];
a[n_] := g[n, n, {n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 29 2017, translated from Maple *)

Formula

a(n) = A047874(2n,n) = A126065(2n,n).
a(n) ~ 16^n * (n-1)! / (Pi * exp(2)). - Vaclav Kotesovec, Mar 27 2016

A293128 Number of standard Young tableaux of 2n cells and height <= n.

Original entry on oeis.org

1, 1, 6, 51, 588, 7990, 126060, 2242618, 44546320, 977152266, 23500234512, 615372604033, 17442275104496, 532242021137346, 17399782340548920, 606732491690590816, 22477989291826848000, 881635273413199806994, 36493478646922003374096, 1589642562747880936613248
Offset: 0

Views

Author

Alois P. Heinz, Sep 30 2017

Keywords

Comments

Also the number of standard Young tableaux of 2n cells and <= n columns.
Also the number of 2n-length words w over n-ary alphabet {a1,a2,...,an} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,an), where #(z,x) counts the letters x in word z. The a(2) = 6 words of length 4 over alphabet {a,b} are: aaaa, aaab, aaba, abaa, aabb, abab.

Links

Crossrefs

Cf. A182172, A267436.

Programs

  • Maple
    h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]
      `if`(n=0 or i=1, h([l[], 1$n]), add(
               g(n-i*j, i-1, [l[], i$j]), j=0..n/i)):
a:= n-> g(2*n, n, []):
seq(a(n), n=0..15);
  • Mathematica
    h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] < j, 0, 1], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Table[1, {n}]]], Sum[g[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
a[n_] := g[2n, n, {}];
a /@ Range[0, 15] (* Jean-François Alcover, Jan 02 2021, after Alois P. Heinz *)

Formula

a(n) = A182172(2n,n).
Showing 1-3 of 3 results.

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