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

0,11

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.

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

Example

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.

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<i, x^w, 1)*b(n-1, i, subsop(i=l[i]+1, l)), 0),
       i=1..nops(l))+ x^w*b(n-1, nops(l)+1, [l[], 1])))
    end:
T:= n->(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 *)

Crossrefs

Row sums give A000085.

Cf. A000217, A003293, A225616, A247386.

Sequence in context: A047971 A029432 A073426 * A346118 A309797 A199840

Adjacent sequences: A232436 A232437 A232438 * A232440 A232441 A232442

Keyword

nonn,tabf

Author

Joerg Arndt and Alois P. Heinz, Feb 23 2014

Status

approved