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A238750 Number T(n,k) of standard Young tableaux with n cells and largest value n in row k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 3
1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 2, 1, 0, 10, 7, 5, 3, 1, 0, 26, 20, 14, 11, 4, 1, 0, 76, 56, 44, 31, 19, 5, 1, 0, 232, 182, 139, 106, 69, 29, 6, 1, 0, 764, 589, 475, 351, 265, 127, 41, 7, 1, 0, 2620, 2088, 1658, 1303, 971, 583, 209, 55, 8, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
Also the number of ballot sequences of length n having last value k.
Also the number of standard Young tableaux with n cells where the row containing the largest value n has length k.
Also the number of ballot sequences of length n where the last value has multiplicity k.
T(0,0) = 1 by convention.
Columns k=0-2 give: A000007, A000085(n-1), A238124(n-1).
T(2n,n) gives A246731.
Row sums give A000085.
LINKS
Joerg Arndt and Alois P. Heinz, Rows n = 0..60, flattened
Wikipedia, Young tableau
EXAMPLE
The 10 tableaux with n=4 cells sorted by the number of the row containing the largest value 4 are:
:[1 4] [1 2 4] [1 3 4] [1 2 3 4]:[1 2] [1 3] [1 2 3]:[1 2] [1 3]:[1]:
:[2] [3] [2] :[3 4] [2 4] [4] :[3] [2] :[2]:
:[3] : :[4] [4] :[3]:
: : : :[4]:
: --------------1-------------- : --------2-------- : ----3---- : 4 :
Their corresponding ballot sequences are: [1,2,3,1], [1,1,2,1], [1,2,1,1], [1,1,1,1], [1,1,2,2], [1,2,1,2], [1,1,1,2], [1,1,2,3], [1,2,1,3], [1,2,3,4]. Thus row 4 = [0, 4, 3, 2, 1].
Triangle T(n,k) begins:
00: 1;
01: 0, 1;
02: 0, 1, 1;
03: 0, 2, 1, 1;
04: 0, 4, 3, 2, 1;
05: 0, 10, 7, 5, 3, 1;
06: 0, 26, 20, 14, 11, 4, 1;
07: 0, 76, 56, 44, 31, 19, 5, 1;
08: 0, 232, 182, 139, 106, 69, 29, 6, 1;
09: 0, 764, 589, 475, 351, 265, 127, 41, 7, 1;
10: 0, 2620, 2088, 1658, 1303, 971, 583, 209, 55, 8, 1;
MAPLE
h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= proc(l) local n; n:=nops(l); `if`(n=0, 1, add(
`if`(i=n or l[i]>l[i+1], x^i *h(subsop(i=
`if`(i=n and l[n]=1, NULL, l[i]-1), l)), 0), i=1..n))
end:
b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]),
add(b(n-i*j, i-1, [l[], i$j]), j=0..n/i)):
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, n, [])):
seq(T(n), n=0..12);
MATHEMATICA
h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j+
Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, l[[i]]}], {i, n}]];
g[l_] := With[{ n = Length[l]}, If[n == 0, 1, Sum[
If[i == n || l[[i]] > l[[i + 1]], x^i *h[ReplacePart[l, i ->
If[i == n && l[[n]] == 1, Nothing, l[[i]] - 1]]], 0], {i, n}]]];
b[n_, i_, l_] := If[n == 0 || i == 1, g[Join[l, Table[1, {n}]]],
Sum[b[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
T[n_] := CoefficientList[b[n, n, {}], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 27 2021, after Maple code *)
CROSSREFS
Sequence in context: A106234 A238125 A062507 * A131044 A246027 A077875
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt and Alois P. Heinz, Mar 04 2014
STATUS
approved

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Last modified April 16 08:27 EDT 2024. Contains 371698 sequences. (Running on oeis4.)