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A238802 Number T(n,k) of standard Young tableaux with n cells where k is the length of the maximal consecutive sequence 1,2,...,k in the first column; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 9
1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 5, 3, 1, 1, 0, 13, 8, 3, 1, 1, 0, 38, 24, 9, 3, 1, 1, 0, 116, 74, 28, 9, 3, 1, 1, 0, 382, 246, 93, 29, 9, 3, 1, 1, 0, 1310, 848, 321, 98, 29, 9, 3, 1, 1, 0, 4748, 3088, 1168, 350, 99, 29, 9, 3, 1, 1, 0, 17848, 11644, 4404, 1302, 356, 99, 29, 9, 3, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

T(0,0) = 1 by convention.

Also the number of ballot sequences of length n with exactly k fixed points.  The fixed points are in the positions 1,2,...,k.

Row sums give A000085.

Diagonal T(2n,n) gives A238803(n).

Diagonal T(2n+1,n) gives A238803(n+1)-1.

T(n,1) = Sum_{k=2..n} T(n,k) = A000085(n)/2 = A001475(n-1) for n>1.

Columns k=2-8 give: A238977, A238978, A238979, A239116, A239117, A239118, A239119.

Conjecture: Generally, column k is asymptotic to sqrt(2)/(2*(k+1)*(k-1)!) * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))), holds for all k<=10. - Vaclav Kotesovec, Mar 08 2014

LINKS

Joerg Arndt and Alois P. Heinz, Rows n = 0..50, flattened

Wikipedia, Young tableau

EXAMPLE

The 10 tableaux with n=4 cells sorted by the length of the maximal consecutive sequence 1,2,...,k in the first column are:

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

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

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

:                                     :                   :     :[4]:

: -----------------1----------------- : --------2-------- : -3- : 4 :

Their corresponding ballot sequences are:

[1, 1, 2, 3]  ->  1 \

[1, 1, 2, 2]  ->  1  \

[1, 1, 1, 2]  ->  1   } -- 5

[1, 1, 2, 1]  ->  1  /

[1, 1, 1, 1]  ->  1 /

[1, 2, 1, 3]  ->  2 \

[1, 2, 1, 2]  ->  2  } --- 3

[1, 2, 1, 1]  ->  2 /

[1, 2, 3, 1]  ->  3 } ---- 1

[1, 2, 3, 4]  ->  4 } ---- 1

Thus row 4 = [0, 5, 3, 1, 1].

Triangle T(n,k) begins:

00:   1;

01:   0,    1;

02:   0,    1,    1;

03:   0,    2,    1,    1;

04:   0,    5,    3,    1,   1;

05:   0,   13,    8,    3,   1,  1;

06:   0,   38,   24,    9,   3,  1,  1;

07:   0,  116,   74,   28,   9,  3,  1,  1;

08:   0,  382,  246,   93,  29,  9,  3,  1,  1;

09:   0, 1310,  848,  321,  98, 29,  9,  3,  1,  1;

10:   0, 4748, 3088, 1168, 350, 99, 29,  9,  3,  1,  1;

MAPLE

b:= proc(n, l) option remember; `if`(n=0, 1,

       b(n-1, [l[], 1]) +add(`if`(i=1 or l[i-1]>l[i],

       b(n-1, subsop(i=l[i]+1, l)), 0), i=1..nops(l)))

    end:

T:= (n, k)-> `if`(n=k, 1, `if`(k=0, 0, b(n-k-1, [2, 1$(k-1)]))):

seq(seq(T(n, k), k=0..n), n=0..14);

MATHEMATICA

b[n_, l_] := b[n, l] = If[n == 0, 1, b[n-1, Append[l, 1]] + Sum[If[i == 1 || l[[i-1]] > l[[i]], b[n-1, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}]]; T[n_, k_] := If[n == k, 1, If[k == 0, 0, b[n-k-1, Join[{2}, Table[1, {k-1}]]]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-Fran├žois Alcover, Jan 06 2015, translated from Maple *)

CROSSREFS

Sequence in context: A077875 A198237 A122049 * A229892 A064879 A173591

Adjacent sequences:  A238799 A238800 A238801 * A238803 A238804 A238805

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt and Alois P. Heinz, Mar 05 2014

STATUS

approved

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Last modified May 14 19:53 EDT 2021. Contains 343903 sequences. (Running on oeis4.)