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
KEYWORD
nonn,tabf
AUTHOR
Joerg Arndt and Alois P. Heinz, Feb 23 2014
STATUS
approved