A232439 Number T(n,k) of standard Young tableaux with n cells and major index k; triangle T(n,k), n>=0, 0<=k<=n*(n-1)/2, read by rows.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 3, 4, 4, 4, 3, 2, 1, 1, 1, 1, 2, 4, 5, 7, 9, 9, 9, 9, 7, 5, 4, 2, 1, 1, 1, 1, 2, 4, 6, 9, 13, 16, 19, 22, 23, 23, 22, 19, 16, 13, 9, 6, 4, 2, 1, 1, 1, 1, 2, 4, 7, 10, 16, 22, 30, 37, 46, 52, 60, 62, 64, 62

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Feb 23 2014

Keywords

• nonn
• tabf

Comments

Rows are symmetric.

The row beginnings converge to A003293.

T(n,k) is also the number of ballot sequences of length n with k the sum of positions of all ascents, see example.

Examples

			For n=4 the 10 tableaux sorted by major index (sum of descent set) are:
:[1 2 3 4]:[1 3 4]:[1 2] [1 2 4]:[1 4] [1 2 3]:[1 3] [1 3]:[1 2]:[1]:
:         :[2]    :[3 4] [3]    :[2]   [4]    :[2]   [2 4]:[3]  :[2]:
:         :       :             :[3]          :[4]        :[4]  :[3]:
:         :       :             :             :           :     :[4]:
: ---0--- : --1-- : -----2----- : -----3----- : ----4---- : -5- : 6 :
Triangle T(n,k) begins:
1;
1;
1, 1;
1, 1, 1, 1;
1, 1, 2, 2, 2, 1,  1;
1, 1, 2, 3, 4, 4,  4,  3,  2,  1,  1;
1, 1, 2, 4, 5, 7,  9,  9,  9,  9,  7,  5,  4,  2,  1,  1;
1, 1, 2, 4, 6, 9, 13, 16, 19, 22, 23, 23, 22, 19, 16, 13, 9, 6, 4, 2, 1, 1;
The 10 ballot sequences of length 4 are:
##   [ ballot seq] ascent positions  sum
01:  [ 1 1 1 1 ]   (none)            0
02:  [ 1 1 1 2 ]   3                 3
03:  [ 1 1 2 1 ]   2                 2
04:  [ 1 1 2 2 ]   2                 2
05:  [ 1 1 2 3 ]   2 + 3             5
06:  [ 1 2 1 1 ]   1                 1
07:  [ 1 2 1 2 ]   1 + 3             4
08:  [ 1 2 1 3 ]   1 + 3             4
09:  [ 1 2 3 1 ]   1 + 2             3
10:  [ 1 2 3 4 ]   1 + 2 + 3         6
The numbers 2, 3, and 4 appear twice, all others once, so the row four is  1, 1, 2, 2, 2, 1, 1.
		

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 0..40, flattened
• FindStat - Combinatorial Statistic Finder, The charge of the tableau, The inversion number of a standard Young tableau as defined by Haglund and Stevens, The cocharge of a standard tableau
• James Haglund, The q,t-Catalan Numbers and the Space of Diagonal Harmonics, AMS University Lecture Series, vol. 41, 2008.
• Wikipedia, Young tableau

Crossrefs

Row sums give A000085.

Cf. A000217, A003293, A225616, A247386.

Programs

Maple

b:= proc(l, i) option remember; `if`(l=[], 1, expand(add(
      `if`(l[j]>`if`(j=1, 0, l[j-1]), `if`(j=1 and l[j]=1,
       b(subsop(1=NULL, l), j-1), b(subsop(j=l[j]-1, l), j))*
       x^`if`(j>i, add(t, t=l), 0), 0), j=1..nops(l))))
    end:
g:= (n, i, l)-> `if`(n=0 or i=1, b([1$n, l[]], nops(l)+n),
                 add(g(n-i*j, i-1, [i$j, l[]]), j=0..n/i)):
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n$2, [])):
seq(T(n), n=0..10);
# second Maple program (counting ballot sequences):
b:= proc(n, v, l) option remember; local w; w:=add(t, t=l);
      `if`(n<1, 1, expand(add(`if`(i=1 or l[i-1]>l[i],
      `if`(v(p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n-1, 1, [1])):
seq(T(n), n=0..10);

Mathematica

b[l_List, i_] := b[l, i] = If[l == {}, 1, Expand[Sum[ If[l[[j]] > If[j == 1, 0, l[[j-1]]], If[j == 1 && l[[j]] == 1, b[ReplacePart[l, 1 -> Sequence[]], j-1], b[ReplacePart[l, j -> l[[j]]-1], j]]*x^If[j>i, Total[l], 0], 0], {j, 1, Length[l]}]]] ; g[n_, i_, l_List] := g[n, i, l] = If[n == 0 || i == 1, b[Join[Array[1&, n], l], Length[l]+n], Sum[g[n-i*j, i-1, Join[Array[i&, j], l]], {j, 0, n/i}]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][g[n, n, {}]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 14 2015, translated from Maple *)