A238123 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having k largest parts, n >= k >= 0.

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 7, 2, 0, 1, 0, 20, 5, 0, 0, 1, 0, 56, 14, 5, 0, 0, 1, 0, 182, 35, 14, 0, 0, 0, 1, 0, 589, 132, 28, 14, 0, 0, 0, 1, 0, 2088, 399, 90, 42, 0, 0, 0, 0, 1, 0, 7522, 1556, 285, 90, 42, 0, 0, 0, 0, 1, 0, 28820, 5346, 1232, 165, 132, 0, 0, 0, 0, 0, 1

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Feb 21 2014

Keywords

• nonn
• tabl

Comments

Also number of standard Young tableaux with last row of length k.

Examples

			Triangle starts:
00: 1;
01: 0,      1;
02: 0,      1,     1;
03, 0,      3,     0,     1;
04: 0,      7,     2,     0,    1;
05: 0,     20,     5,     0,    0,   1;
06: 0,     56,    14,     5,    0,   0,   1;
07: 0,    182,    35,    14,    0,   0,   0, 1;
08: 0,    589,   132,    28,   14,   0,   0, 0, 1;
09: 0,   2088,   399,    90,   42,   0,   0, 0, 0, 1;
10: 0,   7522,  1556,   285,   90,  42,   0, 0, 0, 0, 1;
11: 0,  28820,  5346,  1232,  165, 132,   0, 0, 0, 0, 0, 1;
12: 0, 113092, 21515,  4378,  737, 297, 132, 0, 0, 0, 0, 0, 1;
13: 0, 464477, 82940, 17082, 3003, 572, 429, 0, 0, 0, 0, 0, 0, 1;
...
The T(6,2)=14 ballot sequences of length 6 with 2 maximal elements are (dots for zeros):
01:  [ . . . . 1 1 ]
02:  [ . . . 1 . 1 ]
03:  [ . . . 1 1 . ]
04:  [ . . 1 . . 1 ]
05:  [ . . 1 . 1 . ]
06:  [ . . 1 1 . . ]
07:  [ . . 1 1 2 2 ]
08:  [ . . 1 2 1 2 ]
09:  [ . 1 . . . 1 ]
10:  [ . 1 . . 1 . ]
11:  [ . 1 . 1 . . ]
12:  [ . 1 . 1 2 2 ]
13:  [ . 1 . 2 1 2 ]
14:  [ . 1 2 . 1 2 ]
The T(8,4)=14 such ballot sequences of length 8 and 4 maximal elements are:
01:  [ . . . . 1 1 1 1 ]
02:  [ . . . 1 . 1 1 1 ]
03:  [ . . . 1 1 . 1 1 ]
04:  [ . . . 1 1 1 . 1 ]
05:  [ . . 1 . . 1 1 1 ]
06:  [ . . 1 . 1 . 1 1 ]
07:  [ . . 1 . 1 1 . 1 ]
08:  [ . . 1 1 . . 1 1 ]
09:  [ . . 1 1 . 1 . 1 ]
10:  [ . 1 . . . 1 1 1 ]
11:  [ . 1 . . 1 . 1 1 ]
12:  [ . 1 . . 1 1 . 1 ]
13:  [ . 1 . 1 . . 1 1 ]
14:  [ . 1 . 1 . 1 . 1 ]
These are the (reversed) Dyck words of semi-length 4.
		

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 0..60, flattened
• Wikipedia, Young tableau

Crossrefs

The terms T(2*n,n) are the Catalan numbers (A000108).

Columns k=0-10 give: A000007, A238124, A244099, A244100, A244101, A244102, A244103, A244104, A244105, A244106, A244107.

Row sums give A000085.

Cf. A026794.

Programs

Maple

b:= proc(n, l) option remember; `if`(n<1, x^l[-1],
      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->`if`(n=0, 1, (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, h([l[], 1$n])*x^`if`(n>0, 1,
       `if`(l=[], 0, l[-1])), g(n, i-1, l)+
       `if`(i>n, 0, g(n-i, i, [l[], i])))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(g(n, n, [])):
seq(T(n), n=0..12);

Mathematica

b[n_, l_List] :=  b[n, l] = If[n<1, x^l[[-1]], 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_] := If[n == 0, 1, 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 *)

PARI

(A238123(n,k)=if(k, vecsum(apply(p->n!/Hook(Vecrev(p)), select(p->p[1]==k,partitions(n,[k,n])))), !n)); Hook(P,h=vector(P[1]),L=P[#P])={prod(i=1, L, h[i]=L-i+1)*prod(i=1,#P-1, my(D=-L+L=P[#P-i]); prod(k=0,L-1,h[L-k]+=min(k,D)+1))} \\  M. F. Hasler, Jun 03 2018