A207542 -id:A207542 - cp's OEIS frontend

A214753 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.

1, 0, 1, 0, 2, 1, 0, 4, 4, 1, 0, 10, 16, 6, 1, 0, 26, 66, 34, 8, 1, 0, 76, 296, 192, 58, 10, 1, 0, 232, 1334, 1134, 406, 88, 12, 1, 0, 764, 6322, 6716, 2918, 730, 124, 14, 1, 0, 2620, 30930, 40872, 20718, 6118, 1186, 166, 16, 1, 0, 9496, 158008, 255308, 149826, 50056, 11310, 1796, 214, 18, 1

Offset: 0

Alois P. Heinz, Aug 02 2012

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,    1;
  0,   4,    4,    1;
  0,  10,   16,    6,   1;
  0,  26,   66,   34,   8,  1;
  0,  76,  296,  192,  58, 10,  1;
  0, 232, 1334, 1134, 406, 88, 12,  1;

Alois P. Heinz, Rows n = 0..20, flattened
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229 [math.CO], 2012
Wikipedia, Young tableau

Columns k=0-10 give: A000007(n), A000085(n) for n>0, A273582, A273583, A273584, A273585, A273586, A273587, A273588, A273589, A273590.
Diagonal and lower diagonal give: A000012, A005843.
Row sums give: A207542.
T(2n,n) gives A273591.
Cf. A215086.

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), [])):
T:= (n, k)-> A(n,k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);

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]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, May 23 2016, after Alois P. Heinz *)

T(n,k) = A215086(n,k) - A215086(n,k-1) for k>0, T(n,0) = A215086(n,0) = A000007(n).

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

Alois P. Heinz, Aug 02 2012

			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, ...

Alois P. Heinz, Antidiagonals n = 0..30, flattened
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229 [math.CO], 2012
Wikipedia, Young tableau

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.

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);

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 *)

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

A323657 Number of strict solid partitions of n.

Original entry on oeis.org

1, 1, 1, 4, 4, 7, 16, 19, 28, 40, 82, 94, 145, 190, 274, 463, 580, 802, 1096, 1486, 1948, 3148, 3811, 5314, 6922, 9394, 11971, 16156, 23044, 28966, 38368, 50002, 65116, 83872, 108706, 137917, 192070, 236242, 308698, 390772, 506935, 633982, 817324, 1018090

Offset: 0

Gus Wiseman, Jan 22 2019

nonn

A strict solid partition is an infinite three-dimensional array of distinct positive integers (and any number of zeros) summing to n such that all one-dimensional sections are strictly decreasing until they become all zeros.

			The a(1) = 1 through a(6) = 16 strict solid partitions, represented as chains of chains of integer partitions:
  ((1))  ((2))  ((3))       ((4))       ((5))       ((6))
                ((21))      ((31))      ((32))      ((42))
                ((2)(1))    ((3)(1))    ((41))      ((51))
                ((2))((1))  ((3))((1))  ((3)(2))    ((321))
                                        ((4)(1))    ((4)(2))
                                        ((3))((2))  ((5)(1))
                                        ((4))((1))  ((31)(2))
                                                    ((32)(1))
                                                    ((4))((2))
                                                    ((5))((1))
                                                    ((31))((2))
                                                    ((3)(2)(1))
                                                    ((32))((1))
                                                    ((3)(1))((2))
                                                    ((3)(2))((1))
                                                    ((3))((2))((1))

Alois P. Heinz, Table of n, a(n) for n = 0..740 (first 401 terms from John Tyler Rascoe)

Cf. A000219, A000293 (solid partitions), A000334, A001970, A002974, A008289, A114736, A117433 (strict plane partitions), A207542, A321662, A323657.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
strplptns[n_]:=Join@@Table[Select[ptnplane[Times@@Prime/@y],And[And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}]
Table[Length[Join@@Table[Select[Tuples[strplptns/@y],And[UnsameQ@@Flatten[#],And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#,{n,n}]&/@#)])]&],{y,IntegerPartitions[n]}]],{n,10}]

a(n) = Sum_{k=1..n} A008289(n,k)*A207542(k) for n > 0. - John Tyler Rascoe, Dec 19 2024

a(21) onwards from John Tyler Rascoe, Dec 19 2024

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

Alois P. Heinz, Aug 03 2012

			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;
  ...

Alois P. Heinz, Rows n = 0..20, flattened
S. B. Ekhad and D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012.
Wikipedia, Young tableau

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

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);

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 *)

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

A379277 Number of solid partitions with multiplicities of parts matching the n-th composition in standard order.

Original entry on oeis.org

1, 3, 3, 6, 9, 6, 9, 13, 21, 24, 33, 13, 21, 24, 33, 24, 48, 57, 84, 51, 93, 90, 135, 24, 48, 57, 84, 51, 93, 90, 135, 48, 102, 144, 213, 138, 258, 252, 387, 111, 228, 282, 426, 219, 417, 408, 633, 48, 102, 144, 213, 138, 258, 252, 387, 111, 228, 282, 426, 219

Offset: 1

John Tyler Rascoe, Dec 19 2024

nonn

			The 5th composition in standard order, (2,1) corresponds to a solid partition with 3 parts (a,b,c) with a = b and a > c. There are 9 ways to arrange these parts into valid a solid partition giving a(5) = 9.

John Tyler Rascoe, Table of n, a(n) for n = 1..1024
John Tyler Rascoe, Python program.

Cf. A000041, A000219, A000293, A066099, A207542.

Python
```
# see links
```

a(2^k) = A000219(k+1).

a(2^k-1) = A207542(k) for k > 0.

A379278 Number of solid partitions of n such that all parts occur with the same multiplicity.

Original entry on oeis.org

1, 1, 4, 10, 20, 31, 97, 105, 228, 466, 657, 953, 2958, 2675, 4884, 11635, 13485, 19136, 58099, 48816, 89138, 219474, 197247, 296097, 1026590, 713425, 1099311, 3386891, 2744274, 3788578, 15225795, 8562311, 13588731, 47251379, 28547765, 43887961, 200572890, 90616026

Offset: 0

John Tyler Rascoe, Dec 19 2024

nonn

			For a(3) = 10 there are 6 arrangements of parts (1,1,1), 3 arrangements of parts (2,1), and 1 arrangement of (3).

John Tyler Rascoe, Python program.

Cf. A000041, A000219, A000293, A207542, A323657.

Python
```
# see links
```

a(27)-a(37) from Bert Dobbelaere, Apr 24 2025

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cp's OEIS Frontend

A214753 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.

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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.

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A323657 Number of strict solid partitions of n.

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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.

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A379277 Number of solid partitions with multiplicities of parts matching the n-th composition in standard order.

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Python

Formula

A379278 Number of solid partitions of n such that all parts occur with the same multiplicity.

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Python

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