A238750 Number T(n,k) of standard Young tableaux with n cells and largest value n in row k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 2, 1, 0, 10, 7, 5, 3, 1, 0, 26, 20, 14, 11, 4, 1, 0, 76, 56, 44, 31, 19, 5, 1, 0, 232, 182, 139, 106, 69, 29, 6, 1, 0, 764, 589, 475, 351, 265, 127, 41, 7, 1, 0, 2620, 2088, 1658, 1303, 971, 583, 209, 55, 8, 1

Offset

0,8

Comments

Also the number of ballot sequences of length n having last value k.

Also the number of standard Young tableaux with n cells where the row containing the largest value n has length k.

Also the number of ballot sequences of length n where the last value has multiplicity k.

T(0,0) = 1 by convention.

Columns k=0-2 give: A000007, A000085(n-1), A238124(n-1).

T(2n,n) gives A246731.

Row sums give A000085.

Links

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

Wikipedia, Young tableau

Example

The 10 tableaux with n=4 cells sorted by the number of the row containing the largest value 4 are:
:[1 4] [1 2 4] [1 3 4] [1 2 3 4]:[1 2] [1 3] [1 2 3]:[1 2] [1 3]:[1]:
:[2]   [3]     [2]              :[3 4] [2 4] [4]    :[3]   [2]  :[2]:
:[3]                            :                   :[4]   [4]  :[3]:
:                               :                   :           :[4]:
: --------------1-------------- : --------2-------- : ----3---- : 4 :
Their corresponding ballot sequences are: [1,2,3,1], [1,1,2,1], [1,2,1,1], [1,1,1,1], [1,1,2,2], [1,2,1,2], [1,1,1,2], [1,1,2,3], [1,2,1,3], [1,2,3,4].  Thus row 4 = [0, 4, 3, 2, 1].
Triangle T(n,k) begins:
00:   1;
01:   0,    1;
02:   0,    1,    1;
03:   0,    2,    1,    1;
04:   0,    4,    3,    2,    1;
05:   0,   10,    7,    5,    3,   1;
06:   0,   26,   20,   14,   11,   4,   1;
07:   0,   76,   56,   44,   31,  19,   5,   1;
08:   0,  232,  182,  139,  106,  69,  29,   6,  1;
09:   0,  764,  589,  475,  351, 265, 127,  41,  7,  1;
10:   0, 2620, 2088, 1658, 1303, 971, 583, 209, 55,  8,  1;

Maple

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(l) local n; n:=nops(l); `if`(n=0, 1, add(
     `if`(i=n or l[i]>l[i+1], x^i *h(subsop(i=
     `if`(i=n and l[n]=1, NULL, l[i]-1), l)), 0), i=1..n))
    end:
b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]),
     add(b(n-i*j, i-1, [l[], i$j]), j=0..n/i)):
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, n, [])):
seq(T(n), n=0..12);

Mathematica

h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j+
     Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, l[[i]]}], {i, n}]];
g[l_] := With[{ n = Length[l]}, If[n == 0, 1, Sum[
     If[i == n || l[[i]] > l[[i + 1]], x^i *h[ReplacePart[l, i ->
     If[i == n && l[[n]] == 1, Nothing, l[[i]] - 1]]], 0], {i, n}]]];
b[n_, i_, l_] := If[n == 0 || i == 1, g[Join[l, Table[1, {n}]]],
     Sum[b[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
T[n_] := CoefficientList[b[n, n, {}], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 27 2021, after Maple code *)

Crossrefs

Sequence in context: A106234 A238125 A062507 * A131044 A246027 A077875

Adjacent sequences: A238747 A238748 A238749 * A238751 A238752 A238753

Keyword

nonn,tabl

Author

Joerg Arndt and Alois P. Heinz, Mar 04 2014

Status

approved