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

A215086 Number A(n,k) of solid standard Young tableaux of n cells and height <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 4, 0, 1, 1, 3, 8, 10, 0, 1, 1, 3, 9, 26, 26, 0, 1, 1, 3, 9, 32, 92, 76, 0, 1, 1, 3, 9, 33, 126, 372, 232, 0, 1, 1, 3, 9, 33, 134, 564, 1566, 764, 0, 1, 1, 3, 9, 33, 135, 622, 2700, 7086, 2620, 0, 1, 1, 3, 9, 33, 135, 632, 3106, 13802, 33550, 9496, 0
Offset: 0

Views

Author

Alois P. Heinz, Aug 02 2012

Keywords

Examples

			Square array A(n,k) begins:
  1,   1,    1,    1,    1,    1,    1,    1, ...
  0,   1,    1,    1,    1,    1,    1,    1, ...
  0,   2,    3,    3,    3,    3,    3,    3, ...
  0,   4,    8,    9,    9,    9,    9,    9, ...
  0,  10,   26,   32,   33,   33,   33,   33, ...
  0,  26,   92,  126,  134,  135,  135,  135, ...
  0,  76,  372,  564,  622,  632,  633,  633, ...
  0, 232, 1566, 2700, 3106, 3194, 3206, 3207, ...

Links

Crossrefs

Columns k=0-10 give: A000007, A000085, A215087, A320180, A320181, A320182, A320183, A320184, A320185, A320186, A320187.
Rows n=0-1 give: A000012, A057427.
Main diagonal gives A207542.
Cf. A214753.

Programs

  • Maple
    b:= proc(n, k, l) option remember; `if`(n=0, 1,
       b(n-1, k, [l[], [1]])+ add(`if`(i=1 or nops(l[i]) `if`(k=0, `if`(n=0, 1, 0), b(n, min(n, k), [])):
seq(seq(A(n, d-n), n=0..d), d=0..10);
  • Mathematica
    b[n_, k_, l_] := b[n, k, l] = If[n==0, 1, b[n-1, k, Append[l, {1}]] + Sum[If[i==1 || Length[l[[i]]] Append[l[[i]], 1]]], 0] + Sum[If[l[[i, j]] ReplacePart[ l[[i]], j -> l[[i, j]]+1]]], 0], {j, 1, Length[l[[i]]]} ], {i, 1, Length[l]}]]; A[n_, k_] := If[k==0, If[n==0, 1, 0], b[n, Min[n, k], {}]]; Table[A[n, d-n], {d, 0, 11}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 26 2017, after Alois P. Heinz *)

Formula

A(n,k) = Sum_{i=0..k} A214753(n,i).

A207542 Number of solid standard Young tableaux with n cells.

Original entry on oeis.org

1, 1, 3, 9, 33, 135, 633, 3207, 17589, 102627, 636033, 4161141, 28680717, 207318273, 1567344549, 12345147705, 101013795753, 856212871761, 7501911705747, 67815650852235, 631574151445665, 6051983918989833, 59605200185016639, 602764245172225251, 6252962956009863363
Offset: 0

Views

Author

Matthew C. Russell, Feb 24 2012

Keywords

Comments

A solid standard Young tableaux (SSYT) with n cells is a way of placing the integers from 1 to n in a 3D Young diagram of a plane partition with the property that the entries increase from left to right, back to front, and bottom to top.
It is also the number of almost topological sequences (ATS) for the set N^3 at depth n with (N=set of nonnegative integers). See Balakrishnan et al. for definition and a proof of the bijection between SSYT and ATS. - Suresh Govindarajan, Mar 02 2012
Also the number of paths from a plane partition to the empty partition by repeated trimming. - Wouter Meeussen, Sep 03 2025

Links

Crossrefs

Rows sums of A214753.
Main diagonal of A215086.
Column k=0 of A215120. - Alois P. Heinz, May 12 2014

Programs

  • Mathematica
    b[n_, k_, L_] := b[n, k, L] = If[n == 0, 1, b[n - 1, k, Append[L, {1}]] + Sum[If[i == 1 || Length[L[[i]]] < Length[L[[i - 1]]], b[n - 1, k, ReplacePart[L, i -> Append[L[[i]], 1]]], 0] + Sum[If[L[[i, j]] < k && (i == 1 || L[[i, j]] < L[[i - 1, j]]) && (j == 1 || L[[i, j]] < L[[i, j - 1]]), b[n - 1, k, ReplacePart[L, i -> ReplacePart[L[[i]], j -> L[[i, j]] + 1]]], 0], {j, 1, Length[L[[i]]]}], {i, 1, Length[L]}]];
A[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[n, Min[n, k], {}]];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
a[n_] := a[n] = Sum[T[n, k], {k, 0, n}];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 28 2022, after Alois P. Heinz in A214753 *)

A215120 Number T(n,k) of solid standard Young tableaux of n cells and height >= k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 9, 9, 5, 1, 33, 33, 23, 7, 1, 135, 135, 109, 43, 9, 1, 633, 633, 557, 261, 69, 11, 1, 3207, 3207, 2975, 1641, 507, 101, 13, 1, 17589, 17589, 16825, 10503, 3787, 869, 139, 15, 1, 102627, 102627, 100007, 69077, 28205, 7487, 1369, 183, 17, 1
Offset: 0

Views

Author

Alois P. Heinz, Aug 03 2012

Keywords

Examples

			Triangle T(n,k) begins:
     1;
     1,    1;
     3,    3,    1;
     9,    9,    5,    1;
    33,   33,   23,    7,   1;
   135,  135,  109,   43,   9,   1;
   633,  633,  557,  261,  69,  11,  1;
  3207, 3207, 2975, 1641, 507, 101, 13,  1;
  ...

Links

Crossrefs

Column k=0 gives: A207542.
Diagonal and lower diagonal give: A000012, A005408.
T(2n,n) gives A385413.
Cf. A214753, A215086.

Programs

  • Maple
    b:= proc(n, k, l) option remember; `if`(n=0, 1,
       b(n-1, k, [l[], [1]])+ add(`if`(i=1 or nops(l[i]) `if`(k=0, `if`(n=0, 1, 0), b(n, min(n, k), [])):
H:= (n, k)-> A(n,k) -`if`(k=0, 0, A(n, k-1)):
T:= proc(n, k) option remember; `if`(k=n, 1, T(n, k+1)+ H(n, k)) end:
seq(seq(T(n, k), k=0..n), n=0..10);
  • Mathematica
    b[n_, k_, L_] := b[n, k, L] = If[n == 0, 1, b[n - 1, k, Append[L, {1}]] + Sum[If[i == 1 || Length[L[[i]]] < Length[L[[i - 1]]], b[n - 1, k, ReplacePart[L, i -> Append[L[[i]], 1]]], 0] + Sum[If[L[[i, j]] < k && (i == 1 || L[[i, j]] < L[[i - 1, j]]) && (j == 1 || L[[i, j]] < L[[i, j - 1]]), b[n - 1, k, ReplacePart[L, i -> ReplacePart[L[[i]], j -> L[[i, j]] + 1]]], 0], {j, 1, Length[L[[i]]]}], {i, 1, Length[L]}]];
A[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[n, Min[n, k], {}]];
H[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
T[n_, n_] = 1;
T[n_, k_] := T[n, k] = T[n, k + 1] + H[n, k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 28 2022, after Alois P. Heinz *)

Formula

T(n,n) = 1, T(n,k) = T(n,k+1) + A214753(n,k) for k

A273582 Number of solid standard Young tableaux of n cells and height two.

Original entry on oeis.org

1, 4, 16, 66, 296, 1334, 6322, 30930, 158008, 837530, 4624462, 26379128, 155536148, 944173842, 5898860754, 37848982610, 249347884600, 1684168485302, 11655678956456, 82532887947582, 597399961515754, 4414814240798968, 33281178413298246, 255687224968704234
Offset: 2

Views

Author

Alois P. Heinz, May 25 2016

Keywords

Links

Crossrefs

Column k=2 of A214753.

A273583 Number of solid standard Young tableaux of n cells and height three.

Original entry on oeis.org

1, 6, 34, 192, 1134, 6716, 40872, 255308, 1646736, 10966948, 75443030, 534853224, 3896868930, 29123747126, 222952505194, 1746931730522, 14000986111688, 114736915809792, 960962428696876, 8221685556988910, 71811228291496646, 639892151737631798
Offset: 3

Views

Author

Alois P. Heinz, May 25 2016

Keywords

Links

Crossrefs

Column k=3 of A214753.

A273584 Number of solid standard Young tableaux of n cells and height four.

Original entry on oeis.org

1, 8, 58, 406, 2918, 20718, 149826, 1101760, 8315184, 64408712, 512934920, 4192369318, 35116467578, 300911466640, 2635363943658, 23572245742164, 215262453066404, 2006391770994272, 19083039726219474, 185146905421778870, 1831674943638953366, 18467803852380612492
Offset: 4

Views

Author

Alois P. Heinz, May 25 2016

Keywords

Links

Crossrefs

Column k=4 of A214753.

A273585 Number of solid standard Young tableaux of n cells and height five.

Original entry on oeis.org

1, 10, 88, 730, 6118, 50056, 413170, 3442930, 29255424, 253838072, 2253901296, 20463487776, 189807311174, 1796551402964, 17341237885160, 170630252165826, 1711128376836476, 17486837759278710, 182094398861966064, 1931781562533747898, 20872575572271906478
Offset: 5

Views

Author

Alois P. Heinz, May 25 2016

Keywords

Links

Crossrefs

Column k=5 of A214753.

A273586 Number of solid standard Young tableaux of n cells and height six.

Original entry on oeis.org

1, 12, 124, 1186, 11310, 104212, 960768, 8881520, 83243786, 792762774, 7693204534, 76057329258, 765792782624, 7847188907046, 81815091467488, 867767607899554, 9363465279720294, 102791356120285178, 1148111902688701322, 13046567129596851746, 150810085730131941438
Offset: 6

Views

Author

Alois P. Heinz, May 25 2016

Keywords

Links

Crossrefs

Column k=6 of A214753.

A273587 Number of solid standard Young tableaux of n cells and height seven.

Original entry on oeis.org

1, 14, 166, 1796, 19166, 195902, 1989222, 20131806, 205509762, 2122093502, 22240817802, 236653051932, 2556840927834, 28040548743090, 312159819857544, 3527927125582748, 40486899056049862, 471911116487767150, 5587797803475074264, 67218738406311961560
Offset: 7

Views

Author

Alois P. Heinz, May 25 2016

Keywords

Links

Crossrefs

Column k=7 of A214753.

A273588 Number of solid standard Young tableaux of n cells and height eight.

Original entry on oeis.org

1, 16, 214, 2582, 30454, 341506, 3781000, 41502740, 457409970, 5078883258, 57033626274, 648178880258, 7458864054446, 86908042007900, 1025620931756786, 12262869399150298, 148612621872353230, 1826179906751341758, 22761392908451630606, 287811790986462659190
Offset: 8

Views

Author

Alois P. Heinz, May 25 2016

Keywords

Links

Crossrefs

Column k=8 of A214753.
Showing 1-10 of 13 results. Next

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