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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]) `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 *) 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.)