A238727 - OEIS

A238727

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

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 1, 2, 7, 0, 0, 1, 3, 8, 14, 0, 0, 1, 4, 11, 19, 41, 0, 0, 1, 7, 19, 34, 64, 107, 0, 0, 1, 11, 32, 62, 119, 202, 337, 0, 0, 1, 21, 64, 131, 248, 418, 671, 1066, 0, 0, 1, 36, 124, 277, 545, 943, 1518, 2361, 3691

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OFFSET

0,6

COMMENTS

T(0,0) = 1 by convention.

Also the number of ballot sequences of length n having the last occurrence of the maximal value at position k.

T(n,3) = A051920(n-3) for n>3.

T(2n,n) gives A246818.

Main diagonal gives A238728.

Row sums give A000085.

LINKS

Joerg Arndt and Alois P. Heinz, Rows n = 0..43, flattened

Wikipedia, Young tableau

EXAMPLE

The 10 tableaux with n=4 cells sorted by largest value in the last row:

:[1 3 4]:[1 4] [1 2 4]:[1] [1 2] [1 3] [1 2 3] [1 2] [1 3] [1 2 3 4]:

:[2] :[2] [3] :[2] [3] [2] [4] [3 4] [2 4] :

: :[3] :[3] [4] [4] :

: : :[4] :

: --2-- : -----3----- : ---------------------4--------------------- :

The 10 ballot sequences of length 4 sorted by the position of the last occurrence of the maximal value:

[1, 2, 1, 1] -> 2 } -- 1

[1, 2, 3, 1] -> 3 \ __ 2

[1, 1, 2, 1] -> 3 /

[1, 2, 3, 4] -> 4 \

[1, 1, 2, 3] -> 4 \

[1, 2, 1, 3] -> 4 \

[1, 1, 1, 2] -> 4 } 7

[1, 1, 2, 2] -> 4 /

[1, 2, 1, 2] -> 4 /

[1, 1, 1, 1] -> 4 /

thus row 4 = [0, 0, 1, 2, 7].

Triangle T(n,k) begins:

00: 1;

01: 0, 1;

02: 0, 0, 2;

03: 0, 0, 1, 3;

04: 0, 0, 1, 2, 7;

05: 0, 0, 1, 3, 8, 14;

06: 0, 0, 1, 4, 11, 19, 41;

07: 0, 0, 1, 7, 19, 34, 64, 107;

08: 0, 0, 1, 11, 32, 62, 119, 202, 337;

09: 0, 0, 1, 21, 64, 131, 248, 418, 671, 1066;

10: 0, 0, 1, 36, 124, 277, 545, 943, 1518, 2361, 3691;

MAPLE

h:= proc(l) option remember; local n, s; n:= nops(l); s:= add(i, i=l);

`if`(n=0, 1, add(`if`(i<n and l[i]>l[i+1], h(subsop(i=l[i]-1, l)),

`if`(i=n, (p->add(coeff(p, x, j)*x^`if`(j<s, s, j), j=0..degree(p)))

(h(subsop(i=`if`(l[i]>1, l[i]-1, [][]), l))), 0)), i=1..n))

end:

g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]),

add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)):

T:= n-> (p->seq(coeff(p, x, i), i=0..n))(g(n$2, [])):

seq(T(n), n=0..12);

MATHEMATICA

h[l_] := h[l] = With[{n = Length[l], s = Total[l]},

If[n == 0, 1, Sum[If[i < n && l[[i]] > l[[i + 1]],

h[ReplacePart[l, i -> l[[i]] - 1]], If[i == n, Function[p,

Sum[Coefficient[p, x, j] x^If[j < s, s, j], {j, 0,

Exponent[p, x]}]][h[ReplacePart[l, i -> If[l[[i]] > 1,

l[[i]] - 1, Nothing]]]], 0]], {i, n}]]];

g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Table[1, {n}]]],

Sum[g[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]];

T[n_] := CoefficientList[g[n, n, {}], x];

Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 27 2021, after Maple code )*

CROSSREFS

Sequence in context: A116489 A166373 A202451 * A056885 A029373 A357645

Adjacent sequences: A238724 A238725 A238726 * A238728 A238729 A238730

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt and Alois P. Heinz, Mar 03 2014

STATUS

approved