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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jan 14 2015, translated from Maple *)

CROSSREFS

Row sums give A000085.

Cf. A000217, A003293, A225616, A247386.

Sequence in context: A047971 A029432 A073426 * A309797 A199840 A126067

Adjacent sequences:  A232436 A232437 A232438 * A232440 A232441 A232442

KEYWORD

nonn,tabf

AUTHOR

Joerg Arndt and Alois P. Heinz, Feb 23 2014

STATUS

approved

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Last modified December 15 14:37 EST 2019. Contains 329999 sequences. (Running on oeis4.)