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A238707
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Number T(n,k) of ballot sequences of length n having difference k between the multiplicities of the smallest and the largest value; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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13
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1, 1, 0, 2, 0, 0, 2, 2, 0, 0, 4, 3, 3, 0, 0, 2, 14, 6, 4, 0, 0, 12, 14, 35, 10, 5, 0, 0, 2, 69, 71, 69, 15, 6, 0, 0, 30, 97, 295, 195, 119, 21, 7, 0, 0, 44, 251, 751, 929, 421, 188, 28, 8, 0, 0, 86, 671, 2326, 3044, 2254, 791, 279, 36, 9, 0, 0, 2, 1847, 6524, 11824, 8999, 4696, 1354, 395, 45, 10, 0, 0
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OFFSET
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0,4
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COMMENTS
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Also the number of standard Young tableaux (SYT) with n cells having difference k between the lengths of the first and the last row.
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LINKS
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EXAMPLE
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For n=4 the 10 ballot sequences of length 4 and differences between the multiplicities of the smallest and the largest value are:
[1, 2, 3, 4] -> 1-1 = 0,
[1, 1, 2, 2] -> 2-2 = 0,
[1, 2, 1, 2] -> 2-2 = 0,
[1, 1, 1, 1] -> 4-4 = 0,
[1, 1, 2, 3] -> 2-1 = 1,
[1, 2, 1, 3] -> 2-1 = 1,
[1, 2, 3, 1] -> 2-1 = 1,
[1, 1, 1, 2] -> 3-1 = 2,
[1, 1, 2, 1] -> 3-1 = 2,
[1, 2, 1, 1] -> 3-1 = 2,
thus row 4 = [4, 3, 3, 0, 0].
The 10 tableaux with 4 cells sorted by the difference between the lengths of the first and the last row are:
:[1] [1 2] [1 3] [1 2 3 4]:[1 2] [1 3] [1 4]:[1 2 3] [1 2 4] [1 3 4]:
:[2] [3 4] [2 4] :[3] [2] [2] :[4] [3] [2] :
:[3] :[4] [4] [3] : :
:[4] : : :
: -----------0----------- : -------1------- : ----------2---------- :
Triangle T(n,k) begins:
00: 1;
01: 1, 0;
02: 2, 0, 0;
03: 2, 2, 0, 0;
04: 4, 3, 3, 0, 0;
05: 2, 14, 6, 4, 0, 0;
06: 12, 14, 35, 10, 5, 0, 0;
07: 2, 69, 71, 69, 15, 6, 0, 0;
08: 30, 97, 295, 195, 119, 21, 7, 0, 0;
09: 44, 251, 751, 929, 421, 188, 28, 8, 0, 0;
10: 86, 671, 2326, 3044, 2254, 791, 279, 36, 9, 0, 0;
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MAPLE
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b:= proc(n, l) option remember; `if`(n<1, x^(l[1]-l[-1]),
expand(b(n-1, [l[], 1])+add(`if`(i=1 or l[i-1]>l[i],
b(n-1, subsop(i=l[i]+1, l)), 0), i=1..nops(l))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n-1, [1])):
seq(T(n), n=0..12);
# second Maple program (counting SYT):
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(n, i, l) `if`(n=0 or i=1, (p->h(p)*x^(`if`(p=[], 0, p[1]-
p[-1])))([l[], 1$n]), add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))
end:
T:= n->(p-> seq(coeff(p, x, i), i=0..n))(g(n, n, [])):
seq(T(n), n=0..12);
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MATHEMATICA
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b[n_, l_List] := b[n, l] = If[n<1, x^(l[[1]] - l[[-1]]), Expand[b[n-1, Append[l, 1]] + Sum[If[i == 1 || l[[i-1]] > l[[i]], b[n-1, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n-1, {1}]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 07 2015, translated from Maple *)
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CROSSREFS
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Columns k=0-10 give: A067228, A244295, A244296, A244297, A244298, A244299, A244300, A244301, A244302, A244303, A244304.
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KEYWORD
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AUTHOR
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STATUS
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approved
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