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A238728 Number of standard Young tableaux with n cells where the largest value n is contained in the last row. 2
1, 1, 2, 3, 7, 14, 41, 107, 337, 1066, 3691, 12962, 49061, 188894, 766845, 3182844, 13758383, 60858842, 278312475, 1301323108, 6258671365, 30742575588, 154785692507, 794888735945, 4173162573277, 22318859784416, 121767607626621, 676010926754742 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(0) = 1 by convention.

Also number of ballot sequences of length n where the last position has a maximal value.

Main diagonal of A238727.

LINKS

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..60

Wikipedia, Young tableau

EXAMPLE

a(4) = 7 counts the following tableaux:

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

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

[3] [4] [4]

[4]

corresponding to the following ballot sequences: [1,2,3,4], [1,1,2,3], [1,2,1,3], [1,1,1,2], [1,1,2,2], [1,2,1,2], [1,1,1,1].

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:= l->`if`(l=[], 1, h(subsop(-1=`if`(l[-1]=1, [][], l[-1]-1), l))):

b:= proc(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)) end:

a:= n-> b(n, n, []):

seq(a(n), n=0..28);

MATHEMATICA

h[l_List] := With[{n=Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j+Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[l_List] := If[l == {}, 1, h[If[Last[l] == 1, Most[l], Append[Most[l], Last[l]-1]]]]; b[n_, i_, l_List] := If[n == 0 || i == 1, g[Join[l, Array[1&, n]]], Sum[b[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]; a[n_] := b[n, n, {}]; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Feb 12 2015, after Maple *)

CROSSREFS

Sequence in context: A274538 A113182 A165433 * A271718 A222626 A006627

Adjacent sequences: A238725 A238726 A238727 * A238729 A238730 A238731

KEYWORD

nonn

AUTHOR

Joerg Arndt and Alois P. Heinz, Mar 03 2014

STATUS

approved

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Last modified December 3 15:03 EST 2022. Contains 358534 sequences. (Running on oeis4.)