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 1-4 of 4 results.

A219274 Number T(n,k) of standard Young tableaux for partitions of n into distinct parts with largest part k; triangle T(n,k), k>=0, k<=n<=k*(k+1)/2, read by columns.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 5, 16, 1, 4, 9, 49, 70, 168, 768, 1, 5, 14, 92, 204, 738, 3300, 7887, 15015, 48048, 292864, 1, 6, 20, 153, 405, 1815, 9460, 28743, 101673, 333905, 1946516, 4934930, 14454726, 34918884, 141892608, 1100742656, 1, 7, 27, 235, 715, 3630, 21307
Offset: 0

Views

Author

Alois P. Heinz, Nov 17 2012

Keywords

Comments

T(n,k) is defined for n,k >= 0. T(n,k) = 0 iff n k*(k+1)/2 = A000217(k). The triangle contains only the nonzero terms.

Examples

			T(3,2) = 2:
+------+  +------+
| 1  2 |  | 1  3 |
| 3 .--+  | 2 .--+
+---+     +---+
Triangle T(n,k) begins:
1;
.  1;
.     1;
.     2,  1;
.         3,   1;
.         5,   4,   1;
.        16,   9,   5,   1;
.             49,  14,   6,   1;
.             70,  92,  20,   7,  1;
.            168, 204, 153,  27,  8, 1;
.            768, 738, 405, 235, 35, 9, 1;

Links

Crossrefs

Column heights are A000124(k-1) for k>0.
Column sums give: A219275.
Row sums give: A218293.
Diagonal and lower diagonals give: A000012, A000027 (for n>1), A000096(n-1) (for n>2).
Leftmost nonzero elements give A219339.
Column of leftmost nonzero element is A002024(n) for n>0.
Triangle read by rows reversed gives: A219356.
T(A000217(n),n) = A005118(n+1).

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) local s; s:=i*(i+1)/2;
      `if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
       g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
    end:
T:= (n, k)-> `if`(k>n, 0, g(n-k, k-1, [k])):
seq(seq(T(n, k), n=k..k*(k+1)/2), k=0..7);
  • Mathematica
    h[l_] := Module[{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_] := Module[{s = i(i + 1)/2}, If[n == s, h[Join[l, Table[i - j, {j, 0, i - 1}]]], If[n > s, 0, g[n, i - 1, l] + If[i > n, 0, g[n - i, i - 1, Append[l, i]]]]]];
T[n_, k_] := If[k > n, 0, g[n - k, k - 1, {k}]];
Table[Table[T[n, k], {n, k, k(k + 1)/2}], {k, 0, 7}] // Flatten (* Jean-François Alcover, Sep 01 2023, after Alois P. Heinz *)

Formula

T(n,k) = A219272(n,k) - A219272(n,k-1) for k>0.

A219272 Number A(n,k) of standard Young tableaux for partitions of n into distinct parts with largest part <= k; triangle A(n,k), k>=0, 0<=n<=k*(k+1)/2, read by columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 5, 16, 1, 1, 1, 3, 4, 9, 25, 49, 70, 168, 768, 1, 1, 1, 3, 4, 10, 30, 63, 162, 372, 1506, 3300, 7887, 15015, 48048, 292864, 1, 1, 1, 3, 4, 10, 31, 69, 182, 525, 1911, 5115, 17347, 43758, 149721, 626769, 1946516, 4934930
Offset: 0

Views

Author

Alois P. Heinz, Nov 17 2012

Keywords

Comments

A(n,k) is defined for n,k >= 0. A(n,k) = 0 iff n > k*(k+1)/2 = A000217(k). The triangle contains only the nonzero terms. A(n,k) = A(n,n) for k>=n.

Examples

			A(3,2) = 2:
+------+  +------+
| 1  2 |  | 1  3 |
| 3 .--+  | 2 .--+
+---+     +---+
A(3,3) = 3:
+------+  +------+  +---------+
| 1  2 |  | 1  3 |  | 1  2  3 |
| 3 .--+  | 2 .--+  +---------+
+---+     +---+
Triangle A(n,k) begins:
1,  1,  1,  1,   1,    1,    1,    1,    1, ...
.   1,  1,  1,   1,    1,    1,    1,    1, ...
.       1,  1,   1,    1,    1,    1,    1, ...
.       2,  3,   3,    3,    3,    3,    3, ...
.           3,   4,    4,    4,    4,    4, ...
.           5,   9,   10,   10,   10,   10, ...
.          16,  25,   30,   31,   31,   31, ...
.               49,   63,   69,   70,   70, ...
.               70,  162,  182,  189,  190, ...

Links

Crossrefs

Column heights are A000124.
Column sums give: A219273.
Diagonal gives: A218293.
Leftmost nonzero elements give A219339.
Column of leftmost nonzero element is A002024(n) for n>0.
T(A000217(n),n) = A005118(n+1).

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) local s; s:=i*(i+1)/2;
      `if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
       g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
    end:
A:= (n, k)-> g(n, k, []):
seq(seq(A(n, k), n=0..k*(k+1)/2), k=0..7);
  • 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] = With[{s=i*(i+1)/2}, If[n==s, h[Join[l, Table[ i-j, {j, 0, i-1}]]], If[n>s, 0, g[n, i-1, l] + If[i>n, 0, g[n-i, i-1, Append[l, i]]]]]];
A[n_, k_] := g[n, k, {}];
Table[Table[A[n, k], {n, 0, k*(k+1)/2}], {k, 0, 7}] // Flatten (* Jean-François Alcover, Feb 29 2016, after Alois P. Heinz *)

Formula

T(n,k) = Sum_{i=0..k} A219274(n,i).

A219347 Number of partitions of n into distinct parts with smallest possible largest part.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 4, 3, 2, 2, 1, 1, 1, 5, 4, 3, 2, 2, 1, 1, 1, 6, 5, 4, 3, 2, 2, 1, 1, 1, 8, 6, 5, 4, 3, 2, 2, 1, 1, 1, 10, 8, 6, 5, 4, 3, 2, 2, 1, 1, 1, 12, 10, 8, 6, 5, 4, 3, 2, 2, 1, 1, 1, 15, 12, 10, 8, 6, 5
Offset: 0

Views

Author

Alois P. Heinz, Nov 18 2012

Keywords

Comments

Size of the smallest possible largest part is floor(sqrt(2*n)+1/2) = A002024(n). Records occur at 0, 7, and A000124(k) for k>=5.

Examples

			a(0) = 1: [].
a(7) = 2: [4,2,1], [4,3].
a(16) = 3: [6,4,3,2,1], [6,5,3,2], [6,5,4,1].
a(22) = 4: [7,5,4,3,2,1], [7,6,4,3,2], [7,6,5,3,1], [7,6,5,4].

Links

Crossrefs

Cf. A000009 (records), A219339.

Programs

  • Maple
    g:= proc(n, i) option remember; local s; s:=i*(i+1)/2;
      `if`(n=s, 1, `if`(n>s, 0, g(n, i-1)+ `if`(i>n, 0, g(n-i, i-1))))
    end:
a:= n-> g(n, floor(sqrt(2*n)+1/2)):
seq (a(n), n=0..120);
  • Mathematica
    g[n_, i_] := g[n, i] = Module[{s = i(i+1)/2}, If[n == s, 1, If[n > s, 0, g[n, i - 1] + If[i > n, 0, g[n - i, i - 1]]]]];
a[n_] := g[n, Floor[Sqrt[2n] + 1/2]];
a /@ Range[0, 120] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

A219356 Triangle read by rows: A219274 with rows reversed.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 4, 5, 1, 5, 9, 16, 1, 6, 14, 49, 1, 7, 20, 92, 70, 1, 8, 27, 153, 204, 168, 1, 9, 35, 235, 405, 738, 768, 1, 10, 44, 341, 715, 1815, 3300, 1, 11, 54, 474, 1166, 3630, 9460, 7887, 1, 12, 65, 637, 1794, 6578, 21307, 28743, 15015
Offset: 0

Views

Author

Alois P. Heinz, Nov 18 2012

Keywords

Comments

For more information see A219274.

Examples

			A219274 with rows reversed begins:
  1;
  1;
  1;
  1,  2;
  1,  3;
  1,  4,  5;
  1,  5,  9,  16;
  1,  6, 14,  49;
  1,  7, 20,  92,  70;
  1,  8, 27, 153, 204, 168;
  1,  9, 35, 235, 405, 738, 768;
  ...

Links

Crossrefs

Row lengths are A122797 (for n>0).
Row sums give: A218293.
Last elements of rows give: A219339.

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) local s; s:=i*(i+1)/2;
      `if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
       g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
    end:
T:= (n, k)-> `if`(k>n, 0, g(n-k, k-1, [k])):
seq(seq(T(n, n-k), k=0..(n-floor(sqrt(2*n)+1/2))), n=0..14);
Showing 1-4 of 4 results.

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