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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified December 4 16:41 EST 2022. Contains 358563 sequences. (Running on oeis4.)