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A238123 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having k largest parts, n>=0, 0<=k<=n. 11
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

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

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.

LINKS

Joerg Arndt and Alois P. Heinz, Rows n = 0..60, flattened

Wikipedia, Young tableau

EXAMPLE

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.

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-Fran├žois Alcover, Jan 07 2015, translated from Maple *)

CROSSREFS

Sequence in context: A135481 A180049 A244454 * A128311 A132884 A210473

Adjacent sequences:  A238120 A238121 A238122 * A238124 A238125 A238126

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt and Alois P. Heinz, Feb 21 2014

STATUS

approved

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Last modified May 29 06:46 EDT 2017. Contains 287243 sequences.