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A215120 Number T(n,k) of solid standard Young tableaux of n cells and height >= k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 2
1, 1, 1, 3, 3, 1, 9, 9, 5, 1, 33, 33, 23, 7, 1, 135, 135, 109, 43, 9, 1, 633, 633, 557, 261, 69, 11, 1, 3207, 3207, 2975, 1641, 507, 101, 13, 1, 17589, 17589, 16825, 10503, 3787, 869, 139, 15, 1, 102627, 102627, 100007, 69077, 28205, 7487, 1369, 183, 17, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Rows n = 0..20, flattened

S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012

Wikipedia, Young tableau

FORMULA

T(n,n) = 1, T(n,k) = T(n,k+1) + A214753(n,k) for k<n.

EXAMPLE

Triangle T(n,k) begins:

:    1;

:    1,    1;

:    3,    3,    1;

:    9,    9,    5,    1;

:   33,   33,   23,    7,   1;

:  135,  135,  109,   43,   9,   1;

:  633,  633,  557,  261,  69,  11,  1;

: 3207, 3207, 2975, 1641, 507, 101, 13,  1;

MAPLE

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

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

       b(n-1, k, subsop(i=[l[i][], 1], l)), 0)+ add(`if`(l[i][j]<k and

       (i=1 or l[i][j]<l[i-1][j]) and (j=1 or l[i][j]<l[i][j-1]),

       b(n-1, k, subsop(i=subsop(j=l[i][j]+1, l[i]), l)), 0),

       j=1..nops(l[i])), i=1..nops(l)))

    end:

A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), b(n, min(n, k), [])):

H:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):

T:= proc(n, k) option remember; `if`(k=n, 1, T(n, k+1)+ H(n, k)) end:

seq(seq(T(n, k), k=0..n), n=0..10);

CROSSREFS

Column k=0 gives: A207542.

Diagonal and lower diagonal give: A000012, A005408.

Cf. A214753, A215086.

Sequence in context: A193741 A193824 A108075 * A084145 A122919 A188513

Adjacent sequences:  A215117 A215118 A215119 * A215121 A215122 A215123

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 03 2012

STATUS

approved

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Last modified June 16 07:14 EDT 2019. Contains 324145 sequences. (Running on oeis4.)