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A238128 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having largest descent k, n>=0, 0<=k<=n. 12
1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 5, 4, 1, 0, 0, 7, 13, 5, 1, 0, 0, 11, 37, 21, 6, 1, 0, 0, 15, 100, 78, 31, 7, 1, 0, 0, 22, 265, 292, 133, 43, 8, 1, 0, 0, 30, 694, 1028, 586, 215, 57, 9, 1, 0, 0, 42, 1828, 3691, 2453, 1073, 325, 73, 10, 1, 0, 0, 56, 4815, 13004, 10357, 5058, 1836, 467, 91, 11, 1, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Also number of standard Young tableaux with a pair of cells (v,v+1) such that v lies k rows above v+1, and no pair (u,u+1) with a larger such separation exists.
LINKS
Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..35, flattened
EXAMPLE
Triangle starts:
00: 1;
01: 1, 0;
02: 2, 0, 0;
03: 3, 1, 0, 0;
04: 5, 4, 1, 0, 0;
05: 7, 13, 5, 1, 0, 0;
06: 11, 37, 21, 6, 1, 0, 0;
07: 15, 100, 78, 31, 7, 1, 0, 0;
08: 22, 265, 292, 133, 43, 8, 1, 0, 0;
09: 30, 694, 1028, 586, 215, 57, 9, 1, 0, 0;
10: 42, 1828, 3691, 2453, 1073, 325, 73, 10, 1, 0, 0;
11: 56, 4815, 13004, 10357, 5058, 1836, 467, 91, 11, 1, 0, 0;
12: 77, 12867, 46452, 43462, 23953, 9631, 2941, 645, 111, 12, 1, 0, 0;
...
MAPLE
b:= proc(n, v, l) option remember; `if`(n<1, 1, expand(add(
`if`(i=1 or l[i-1]>l[i], (p->`if`(i<v, add(coeff(p, x, h)*
`if`(h<v-i, x^(v-i), x^h), h=0..max(v-i, degree(p))), p))
(b(n-1, i, subsop(i=l[i]+1, l))), 0), i=1..nops(l))+
b(n-1, nops(l)+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..12);
MATHEMATICA
b[n_, v_, l_List] := b[n, v, l] = If[n<1, 1, Expand[Sum[If[i == 1 || l[[i-1]] > l[[i]], Function[{p}, If[i<v, Sum[Coefficient[p, x, h]* If[h < v-i, x^(v-i), x^h], {h, 0, Max[v-i, Exponent[p, x]]}], p]][b[n-1, i, ReplacePart[l, i -> l[[i]]+1]]], 0], {i, 1, Length[l]}] + b[n-1, Length[l]+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, 12}] // Flatten (* Jean-François Alcover, Jan 07 2015, translated from Maple *)
CROSSREFS
Row sums are A000085.
Cf. A238129.
Sequence in context: A369312 A259479 A238343 * A238121 A171380 A323592
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt and Alois P. Heinz, Feb 21 2014
STATUS
approved

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Last modified March 18 22:50 EDT 2024. Contains 370951 sequences. (Running on oeis4.)