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 1-10 of 13 results. Next

A219311 Number T(n,k) of standard Young tableaux for partitions of n into exactly k distinct parts; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 3, 0, 1, 9, 0, 1, 14, 16, 0, 1, 34, 35, 0, 1, 55, 134, 0, 1, 125, 435, 0, 1, 209, 1213, 768, 0, 1, 461, 3454, 2310, 0, 1, 791, 10484, 11407, 0, 1, 1715, 28249, 44187, 0, 1, 3002, 80302, 200044, 0, 1, 6434, 231895, 680160, 292864
Offset: 0

Views

Author

Alois P. Heinz, Nov 17 2012

Keywords

Comments

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=A003056(n). T(n,k) = 0 for k>A003056(n).

Examples

			A(4,2) = 3:
  +---------+  +---------+  +---------+
  | 1  2  3 |  | 1  2  4 |  | 1  3  4 |
  | 4 .-----+  | 3 .-----+  | 2 .-----+
  +---+        +---+        +---+
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1;
  0,  1,    2;
  0,  1,    3;
  0,  1,    9;
  0,  1,   14,    16;
  0,  1,   34,    35;
  0,  1,   55,   134;
  0,  1,  125,   435;
  0,  1,  209,  1213,   768;
  0,  1,  461,  3454,  2310;
  0,  1,  791, 10484, 11407;
  ...

Links

Crossrefs

Columns k=0-10 give: A000007, A000012 (for n>0), A047171(n) = A037952(n)-1, A219316, A219317, A219318, A219319, A219320, A219321, A219322, A219323.
Row sums give: A218293.
Row lengths are 1 + A003056(n).
T(A000217(k),k) = A005118(k+1).
Cf. A219272, A219274.

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, k, l) `if`(n=0, h(l), `if`(n>k*(i-(k-1)/2), 0,
      g(n, i-1, min(k, i-1), l)+`if`(i>n, 0, g(n-i, i-1, k-1, [l[], i]))))
    end:
A:= proc(n, k) option remember; `if`(k<0, 0, g(n, n, k, [])) end:
T:= (n, k)-> A(n, k) -A(n, k-1):
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..20);
  • Mathematica
    h[l_] := With[{n = Length[l]}, Sum[i, {i, 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_, k_, l_] := If[n == 0, h[l], If[n > k*(i-(k-1)/2), 0, g[n, i-1, Min[k, i-1], l] + If[i > n, 0, g[n-i, i-1, k-1, Append[l, i]]]]];
a[n_, k_] := a[n, k] = If[k < 0, 0, g[n, n, k, {}]];
t[n_, k_] := a[n, k] - a[n, k-1];
Table[Table[t[n, k], {k, 0, Floor[(Sqrt[1+8*n]-1)/2]}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

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).

A219320 Number of standard Young tableaux for partitions of n into exactly 7 distinct parts.

Original entry on oeis.org

48608795688960, 295284192952320, 2741894304901440, 18535141513347030, 134524383564933720, 851007098153745060, 5822391651578231460, 37395948352954386420, 238518115727229867660, 1501480486903096567740, 9413700760748972005500, 58167406634979463024710
Offset: 28

Views

Author

Alois P. Heinz, Nov 17 2012

Keywords

Links

Crossrefs

Column k=7 of A219311.
Cf. A218293.

A219316 Number of standard Young tableaux for partitions of n into exactly 3 distinct parts.

Original entry on oeis.org

16, 35, 134, 435, 1213, 3454, 10484, 28249, 80302, 231895, 638406, 1798515, 5170279, 14361074, 40675562, 116701060, 327587324, 931854890, 2678822398, 7577813175, 21658478151, 62401989636, 177658786252, 509822342794, 1472491312385, 4213745453731, 12134760359950
Offset: 6

Views

Author

Alois P. Heinz, Nov 17 2012

Keywords

Links

Crossrefs

Column k=3 of A219311.
Cf. A218293.

Formula

a(n) ~ 3^(n+3/2) / (2*sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 31 2014

A219317 Number of standard Young tableaux for partitions of n into exactly 4 distinct parts.

Original entry on oeis.org

768, 2310, 11407, 44187, 200044, 680160, 2769674, 9826918, 38483206, 135059866, 515249581, 1829452107, 6941537898, 24678730371, 92755537994, 333149285650, 1252530682570, 4513808634840, 16936935284163, 61508180909442, 231189178986445, 843098892380280
Offset: 10

Views

Author

Alois P. Heinz, Nov 17 2012

Keywords

Links

Crossrefs

Column k=4 of A219311.
Cf. A218293.

Formula

a(n) ~ 2^(2*n+5) / (Pi*n^3). - Vaclav Kotesovec, Sep 13 2014

A219318 Number of standard Young tableaux for partitions of n into exactly 5 distinct parts.

Original entry on oeis.org

292864, 1153152, 7194434, 33888582, 177959434, 861962968, 4036054898, 18519351642, 85808400115, 389017226948, 1778061013340, 7967135309510, 35973133665285, 161398383117645, 726152765571840, 3256479005867430, 14629885404315411, 65641088599945380
Offset: 15

Views

Author

Alois P. Heinz, Nov 17 2012

Keywords

Links

Crossrefs

Column k=5 of A219311.
Cf. A218293.

A219319 Number of standard Young tableaux for partitions of n into exactly 6 distinct parts.

Original entry on oeis.org

1100742656, 5462865408, 42035926724, 238839304110, 1477773782690, 8119282473120, 49406279584740, 259405071568305, 1468158383705685, 7798557001165665, 42744495396935010, 224697430361576340, 1226112009886575180, 6397760480647576200, 34422065224987469772
Offset: 21

Views

Author

Alois P. Heinz, Nov 17 2012

Keywords

Links

Crossrefs

Column k=6 of A219311.
Cf. A218293.

A219321 Number of standard Young tableaux for partitions of n into exactly 8 distinct parts.

Original entry on oeis.org

29258366996258488320, 212593716124699852800, 2338077979922915527680, 18541315347775500731880, 156386347073221236234900, 1136852065645214098726260, 8834800018708598317055880, 63128042819798223843289680, 473458147812239316816345390, 3325190851643455231559076330
Offset: 36

Views

Author

Alois P. Heinz, Nov 17 2012

Keywords

Links

Crossrefs

Column k=8 of A219311.
Cf. A218293.

A219322 Number of standard Young tableaux for partitions of n into exactly 9 distinct parts.

Original entry on oeis.org

273035280663535522487992320, 2329440559042398325938585600, 29867922172910654180714311680, 274273837545154589560694664960, 2661165012109500556315841832780, 22087643957707583932955480283900, 194567964473214023234175600529800, 1559754564113482833062794519391700
Offset: 45

Views

Author

Alois P. Heinz, Nov 17 2012

Keywords

Links

Crossrefs

Column k=9 of A219311.
Cf. A218293.
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