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A238129
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Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having largest ascent k, n>=0, 0<=k<=n.
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12
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1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 8, 1, 0, 0, 1, 19, 5, 1, 0, 0, 1, 47, 21, 6, 1, 0, 0, 1, 114, 78, 31, 7, 1, 0, 0, 1, 286, 292, 133, 43, 8, 1, 0, 0, 1, 723, 1028, 586, 215, 57, 9, 1, 0, 0, 1, 1869, 3691, 2453, 1073, 325, 73, 10, 1, 0, 0, 1, 4870, 13004, 10357, 5058, 1836, 467, 91, 11, 1, 0, 0
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OFFSET
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0,8
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COMMENTS
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Also number of standard Young tableaux with a pair of cells (v,v+1) such that v lies k rows below v+1, and no pair (u,u+1) with a larger such separation exists.
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LINKS
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EXAMPLE
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Triangle starts:
00: 1;
01: 1, 0;
02: 1, 1, 0;
03: 1, 3, 0, 0;
04: 1, 8, 1, 0, 0;
05: 1, 19, 5, 1, 0, 0;
06: 1, 47, 21, 6, 1, 0, 0;
07: 1, 114, 78, 31, 7, 1, 0, 0;
08: 1, 286, 292, 133, 43, 8, 1, 0, 0;
09: 1, 723, 1028, 586, 215, 57, 9, 1, 0, 0;
10: 1, 1869, 3691, 2453, 1073, 325, 73, 10, 1, 0, 0;
11: 1, 4870, 13004, 10357, 5058, 1836, 467, 91, 11, 1, 0, 0;
12: 1, 12943, 46452, 43462, 23953, 9631, 2941, 645, 111, 12, 1, 0, 0;
...
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MAPLE
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b:= proc(n, v, l) option remember; local m; m:=nops(l);
`if`(n<1, 1, expand(add(`if`(i=1 or l[i-1]>l[i],
(p->`if`(v<i, add(coeff(p, x, h)*`if`(h<i-v,
x^(i-v), x^h), h=0..max(i-v, degree(p))), p))
(b(n-1, i, subsop(i=l[i]+1, l))), 0), i=1..m)+
(p->add(coeff(p, x, h)*`if`(h<m+1-v, x^(m+1-v), x^h),
h=0..max(m+1-v, degree(p))))(b(n-1, m+1, [l[], 1]))))
end:
T:= n->(p->seq(coeff(p, x, i), i=0..n))(b(n-1, 1, [1])):
seq(T(n), n=0..14);
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MATHEMATICA
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b[n_, v_, l_List] := b[n, v, l] = Module[{m = Length[l]}, If[n<1, 1, Expand[Sum[If[i == 1 || l[[i-1]]>l[[i]], Function[{p}, If[v<i, Sum[Coefficient[p, x, h]*If[h<i-v, x^(i-v), x^h], {h, 0, Max[i-v, Exponent[p, x]]}], p]][b[n-1, i, ReplacePart[l, i -> l[[i]]+1]]], 0], {i, 1, m}] + Function[{p}, Sum[Coefficient[p, x, h]*If[h<m+1-v, x^(m+1-v), x^h], {h, 0, Max[m+1-v, Exponent[p, x]]}]][b[n-1, m+1, Append[l, 1]]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n-1, 1, {1}]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 07 2015, translated from Maple *)
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CROSSREFS
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Columns k=0-10 give: A000012, A244208, A244198, A244199, A244200, A244201, A244202, A244203, A244204, A244205, A244206.
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KEYWORD
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AUTHOR
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STATUS
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approved
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