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A215204 Number A(n,k) of solid standard Young tableaux of cylindrical shape lambda X k, where lambda ranges over all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals. 6
1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 4, 5, 1, 1, 10, 26, 10, 7, 1, 1, 28, 276, 258, 26, 11, 1, 1, 84, 3740, 14318, 3346, 76, 15, 1, 1, 264, 58604, 1161678, 1214358, 54108, 232, 22, 1, 1, 858, 1010616, 118316062, 741215012, 150910592, 1054256, 764, 30 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
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
EXAMPLE
Square array A(n,k) begins:
: 1, 1, 1, 1, 1, 1, ...
: 1, 1, 1, 1, 1, 1, ...
: 2, 2, 4, 10, 28, 84, ...
: 3, 4, 26, 276, 3740, 58604, ...
: 5, 10, 258, 14318, 1161678, 118316062, ...
: 7, 26, 3346, 1214358, 741215012, 620383261034, ...
: 11, 76, 54108, 150910592, 840790914296, 7137345113624878, ...
MAPLE
b:= proc(l) option remember; local m; m:= nops(l);
`if`({map(x-> x[], l)[]}minus{0}={}, 1, add(add(`if`(l[i][j]>
`if`(i=m or nops(l[i+1])<j, 0, l[i+1][j]) and l[i][j]>
`if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
end:
g:= proc(n, i, k, l) `if`(n=0 or i=1, b(map(x-> [k$x], [l[], 1$n])),
add(g(n-i*j, i-1, k, [l[], i$j]), j=0..n/i))
end:
A:= (n, k)-> g(n, n, k, []):
seq(seq(A(n, d-n), n=0..d), d=0..10);
MATHEMATICA
b[l_] := b[l] = With[{m = Length[l]}, If[Union[l // Flatten] ~Complement~ {0} == {}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i + 1]]] < j, 0, l[[i + 1, j]]] && l[[i, j]] > If[Length[l[[i]]] == j, 0, l[[i, j + 1]]], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]] - 1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, m}]]];
g[n_, i_, k_, l_] := If[n == 0 || i == 1, b[Table[k, {#}]& /@ Join[l, Table[1, {n}]]], Sum[g[n - i*j, i - 1, k, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
A[n_, k_] := g[n, n, k, {}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Sep 24 2022, after Alois P. Heinz *)
CROSSREFS
Columns k=0-5 give: A000041, A000085, A215266, A290202, A290214, A290274.
Rows n=0+1, 2-5 give: A000012, 2*A000108, 2*A005789 + A006335, 2*A005790 + 2*A213978 + A114714, 2*A005791 + 2*A215220 + 2*A213932 + A214638.
Main diagonal gives A290225.
Sequence in context: A354273 A282748 A145972 * A260876 A152650 A184219
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 05 2012
STATUS
approved

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Last modified March 29 01:36 EDT 2024. Contains 371264 sequences. (Running on oeis4.)