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A238121 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having exactly k descents, n>=0, 0<=k<=n. 15
1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 5, 5, 0, 0, 0, 7, 16, 3, 0, 0, 0, 11, 43, 21, 1, 0, 0, 0, 15, 99, 101, 17, 0, 0, 0, 0, 22, 215, 373, 145, 9, 0, 0, 0, 0, 30, 430, 1174, 836, 146, 4, 0, 0, 0, 0, 42, 834, 3337, 3846, 1324, 112, 1, 0, 0, 0, 0, 56, 1529, 8642, 15002, 8786, 1615, 66, 0, 0, 0, 0, 0, 77, 2765, 21148, 52132, 47013, 15403, 1582, 32, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Also number of standard Young tableaux such that there are k pairs of cells (v,v+1) with v+1 lying in a row above v.
T(2n,n) gives A241804.
T(2n+1,n) gives A241805.
Row sums are A000085.
T(n*(n+1)/2,n*(n-1)/2) = 1.
A238122 is another version with zeros omitted.
LINKS
Joerg Arndt and Alois P. Heinz, Rows n = 0..50, flattened
EXAMPLE
Triangle starts:
1;
1, 0;
2, 0, 0;
3, 1, 0, 0;
5, 5, 0, 0, 0;
7, 16, 3, 0, 0, 0;
11, 43, 21, 1, 0, 0, 0;
15, 99, 101, 17, 0, 0, 0, 0;
22, 215, 373, 145, 9, 0, 0, 0, 0;
30, 430, 1174, 836, 146, 4, 0, 0, 0, 0;
42, 834, 3337, 3846, 1324, 112, 1, 0, 0, 0, 0;
56, 1529, 8642, 15002, 8786, 1615, 66, 0, 0, 0, 0, 0;
77, 2765, 21148, 52132, 47013, 15403, 1582, 32, 0, 0, 0, 0, 0;
101, 4792, 48713, 164576, 214997, 112106, 21895, 1310, 14, 0, 0, 0, 0, 0;
...
The T(5,1) = 16 ballot sequences of length n=5 with k=1 descent are (dots for zeros):
01: [ . . . 1 . ]
02: [ . . 1 . . ]
03: [ . . 1 . 1 ]
04: [ . . 1 . 2 ]
05: [ . . 1 1 . ]
06: [ . . 1 2 . ]
07: [ . . 1 2 1 ]
08: [ . 1 . . . ]
09: [ . 1 . . 1 ]
10: [ . 1 . . 2 ]
11: [ . 1 . 1 2 ]
12: [ . 1 . 2 3 ]
13: [ . 1 2 . . ]
14: [ . 1 2 . 1 ]
15: [ . 1 2 . 3 ]
16: [ . 1 2 3 . ]
MAPLE
b:= proc(n, v, l) option remember; `if`(n<1, 1, expand(
add(`if`(i=1 or l[i-1]>l[i], `if`(i<v, x, 1)*
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..14);
MATHEMATICA
b[n_, v_, l_] := b[n, v, l] = If[n<1, 1, Sum[If[i == 1 || l[[i-1]] > l[[i]], If[i<v, x, 1]*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, 14}] // Flatten (* Jean-François Alcover, Jan 06 2015, translated from Maple *)
CROSSREFS
Sequence in context: A259479 A238343 A238128 * A171380 A323592 A170980
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt and Alois P. Heinz, Feb 21 2014
STATUS
approved

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Last modified March 19 06:32 EDT 2024. Contains 370953 sequences. (Running on oeis4.)