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A238707 Number T(n,k) of ballot sequences of length n having difference k between the multiplicities of the smallest and the largest value; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 13
1, 1, 0, 2, 0, 0, 2, 2, 0, 0, 4, 3, 3, 0, 0, 2, 14, 6, 4, 0, 0, 12, 14, 35, 10, 5, 0, 0, 2, 69, 71, 69, 15, 6, 0, 0, 30, 97, 295, 195, 119, 21, 7, 0, 0, 44, 251, 751, 929, 421, 188, 28, 8, 0, 0, 86, 671, 2326, 3044, 2254, 791, 279, 36, 9, 0, 0, 2, 1847, 6524, 11824, 8999, 4696, 1354, 395, 45, 10, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Also the number of standard Young tableaux (SYT) with n cells having difference k between the lengths of the first and the last row.

LINKS

Joerg Arndt and Alois P. Heinz, Rows n = 0..67, flattened

Wikipedia, Young tableau

EXAMPLE

For n=4 the 10 ballot sequences of length 4 and differences between the multiplicities of the smallest and the largest value are:

[1, 2, 3, 4]  ->  1-1 = 0,

[1, 1, 2, 2]  ->  2-2 = 0,

[1, 2, 1, 2]  ->  2-2 = 0,

[1, 1, 1, 1]  ->  4-4 = 0,

[1, 1, 2, 3]  ->  2-1 = 1,

[1, 2, 1, 3]  ->  2-1 = 1,

[1, 2, 3, 1]  ->  2-1 = 1,

[1, 1, 1, 2]  ->  3-1 = 2,

[1, 1, 2, 1]  ->  3-1 = 2,

[1, 2, 1, 1]  ->  3-1 = 2,

thus row 4 = [4, 3, 3, 0, 0].

The 10 tableaux with 4 cells sorted by the difference between the lengths of the first and the last row are:

:[1] [1 2] [1 3] [1 2 3 4]:[1 2] [1 3] [1 4]:[1 2 3] [1 2 4] [1 3 4]:

:[2] [3 4] [2 4]          :[3]   [2]   [2]  :[4]     [3]     [2]    :

:[3]                      :[4]   [4]   [3]  :                       :

:[4]                      :                 :                       :

: -----------0----------- : -------1------- : ----------2---------- :

Triangle T(n,k) begins:

00:   1;

01:   1,   0;

02:   2,   0,    0;

03:   2,   2,    0,    0;

04:   4,   3,    3,    0,    0;

05:   2,  14,    6,    4,    0,   0;

06:  12,  14,   35,   10,    5,   0,   0;

07:   2,  69,   71,   69,   15,   6,   0,  0;

08:  30,  97,  295,  195,  119,  21,   7,  0,  0;

09:  44, 251,  751,  929,  421, 188,  28,  8,  0,  0;

10:  86, 671, 2326, 3044, 2254, 791, 279, 36,  9,  0,  0;

MAPLE

b:= proc(n, l) option remember; `if`(n<1, x^(l[1]-l[-1]),

      expand(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-> (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, (p->h(p)*x^(`if`(p=[], 0, p[1]-

      p[-1])))([l[], 1$n]), add(g(n-i*j, i-1, [l[], i$j]), j=0..n/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]] - l[[-1]]), Expand[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_] := 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

Columns k=0-10 give: A067228, A244295, A244296, A244297, A244298, A244299, A244300, A244301, A244302, A244303, A244304.

T(2n,n) gives A244305.

Row sums give A000085.

Sequence in context: A137676 A333755 A238130 * A181111 A353856 A216800

Adjacent sequences:  A238704 A238705 A238706 * A238708 A238709 A238710

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt and Alois P. Heinz, Mar 03 2014

STATUS

approved

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Last modified August 9 21:08 EDT 2022. Contains 356026 sequences. (Running on oeis4.)