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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. 13
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

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

FORMULA

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

EXAMPLE

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, ...

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])<nops(l[i-1]),

       b(n-1, k, subsop(i=[l[i][], 1], l)), 0)+ add(`if`(l[i][j]<k and

       (i=1 or l[i][j]<l[i-1][j]) and (j=1 or l[i][j]<l[i][j-1]),

       b(n-1, k, subsop(i=subsop(j=l[i][j]+1, l[i]), l)), 0),

       j=1..nops(l[i])), i=1..nops(l)))

    end:

A:= (n, k)-> `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]]]<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], {}]]; Table[A[n, d-n], {d, 0, 11}, {n, 0, d}] // Flatten (* Jean-Fran├žois Alcover, Jan 26 2017, after Alois P. Heinz *)

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.

Sequence in context: A299919 A238270 A292521 * A261440 A295684 A276890

Adjacent sequences:  A215083 A215084 A215085 * A215087 A215088 A215089

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 02 2012

STATUS

approved

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Last modified August 3 08:21 EDT 2021. Contains 346435 sequences. (Running on oeis4.)