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A238727
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Number T(n,k) of standard Young tableaux with n cells where k is the largest value in the last row; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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3
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1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 1, 2, 7, 0, 0, 1, 3, 8, 14, 0, 0, 1, 4, 11, 19, 41, 0, 0, 1, 7, 19, 34, 64, 107, 0, 0, 1, 11, 32, 62, 119, 202, 337, 0, 0, 1, 21, 64, 131, 248, 418, 671, 1066, 0, 0, 1, 36, 124, 277, 545, 943, 1518, 2361, 3691
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OFFSET
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0,6
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COMMENTS
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T(0,0) = 1 by convention.
Also the number of ballot sequences of length n having the last occurrence of the maximal value at position k.
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LINKS
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EXAMPLE
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The 10 tableaux with n=4 cells sorted by largest value in the last row:
:[1 3 4]:[1 4] [1 2 4]:[1] [1 2] [1 3] [1 2 3] [1 2] [1 3] [1 2 3 4]:
:[2] :[2] [3] :[2] [3] [2] [4] [3 4] [2 4] :
: :[3] :[3] [4] [4] :
: : :[4] :
: --2-- : -----3----- : ---------------------4--------------------- :
The 10 ballot sequences of length 4 sorted by the position of the last occurrence of the maximal value:
[1, 2, 1, 1] -> 2 } -- 1
[1, 2, 3, 1] -> 3 \ __ 2
[1, 1, 2, 1] -> 3 /
[1, 2, 3, 4] -> 4 \
[1, 1, 2, 3] -> 4 \
[1, 2, 1, 3] -> 4 \
[1, 1, 1, 2] -> 4 } 7
[1, 1, 2, 2] -> 4 /
[1, 2, 1, 2] -> 4 /
[1, 1, 1, 1] -> 4 /
thus row 4 = [0, 0, 1, 2, 7].
Triangle T(n,k) begins:
00: 1;
01: 0, 1;
02: 0, 0, 2;
03: 0, 0, 1, 3;
04: 0, 0, 1, 2, 7;
05: 0, 0, 1, 3, 8, 14;
06: 0, 0, 1, 4, 11, 19, 41;
07: 0, 0, 1, 7, 19, 34, 64, 107;
08: 0, 0, 1, 11, 32, 62, 119, 202, 337;
09: 0, 0, 1, 21, 64, 131, 248, 418, 671, 1066;
10: 0, 0, 1, 36, 124, 277, 545, 943, 1518, 2361, 3691;
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MAPLE
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h:= proc(l) option remember; local n, s; n:= nops(l); s:= add(i, i=l);
`if`(n=0, 1, add(`if`(i<n and l[i]>l[i+1], h(subsop(i=l[i]-1, l)),
`if`(i=n, (p->add(coeff(p, x, j)*x^`if`(j<s, s, j), j=0..degree(p)))
(h(subsop(i=`if`(l[i]>1, l[i]-1, [][]), l))), 0)), i=1..n))
end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]),
add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)):
T:= n-> (p->seq(coeff(p, x, i), i=0..n))(g(n$2, [])):
seq(T(n), n=0..12);
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MATHEMATICA
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h[l_] := h[l] = With[{n = Length[l], s = Total[l]},
If[n == 0, 1, Sum[If[i < n && l[[i]] > l[[i + 1]],
h[ReplacePart[l, i -> l[[i]] - 1]], If[i == n, Function[p,
Sum[Coefficient[p, x, j] x^If[j < s, s, j], {j, 0,
Exponent[p, x]}]][h[ReplacePart[l, i -> If[l[[i]] > 1,
l[[i]] - 1, Nothing]]]], 0]], {i, n}]]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Table[1, {n}]]],
Sum[g[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
T[n_] := CoefficientList[g[n, n, {}], x];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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