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A214775 Number T(n,k) of solid standard Young tableaux of shape [[n,k],[n-k]]; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 8
1, 1, 1, 2, 6, 2, 5, 25, 25, 5, 14, 98, 174, 98, 14, 42, 378, 962, 962, 378, 42, 132, 1452, 4804, 7020, 4804, 1452, 132, 429, 5577, 22689, 43573, 43573, 22689, 5577, 429, 1430, 21450, 103510, 245962, 325590, 245962, 103510, 21450, 1430 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

T(n,k) is odd if and only if n = 2^i-1 for i in {0, 1, 2, ... } = A001477.

LINKS

Alois P. Heinz, Rows n = 0..140, 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

EXAMPLE

Triangle T(n,k) begins:

:   1;

:   1,    1;

:   2,    6,    2;

:   5,   25,   25,    5;

:  14,   98,  174,   98,   14;

:  42,  378,  962,  962,  378,   42;

: 132, 1452, 4804, 7020, 4804, 1452, 132;

MAPLE

b:= proc(x, y, z) option remember; `if`(z>y, b(x, z, y),

      `if`({x, y, z}={0}, 1, `if`(x>y and x>z, b(x-1, y, z), 0)+

      `if`(y>0, b(x, y-1, z), 0)+ `if`(z>0, b(x, y, z-1), 0)))

    end:

T:= (n, k)-> b(n, k, n-k):

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

MATHEMATICA

b[x_, y_, z_] := b[x, y, z] = If[z>y, b[x, z, y], If[Union[{x, y, z}] == {0}, 1, If[x>y && x>z, b[x-1, y, z], 0] + If[y>0, b[x, y-1, z], 0] + If[z>0, b[x, y, z-1], 0]]]; T[n_, k_] := b[n, k, n-k]; Table[T[n, k] , {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jan 15 2014, translated from Maple *)

PROG

(Sage)

@CachedFunction

def B(x, y, z) :

    if z > y : return B(x, z, y)

    if x==y and y==z and z==0 : return 1

    a = B(x-1, y, z) if x>y and x>z else 0

    b = B(x, y-1, z) if y>0 else 0

    c = B(x, y, z-1) if z>0 else 0

    return a + b + c

T = lambda n, k: B(n, k, n-k)

[[T(n, k) for k in (0..n)] for n in (0..10)]

# After Maple code of Alois P. Heinz. Peter Luschny, Jul 30 2012

CROSSREFS

Columns 0-5 give: A000108, A214955, A215298, A215299, A215300, A215301.

Row sums give: A215002.

Central row elements give: A214801.

Sequence in context: A064850 A151853 A268766 * A196201 A128045 A011325

Adjacent sequences:  A214772 A214773 A214774 * A214776 A214777 A214778

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 28 2012

STATUS

approved

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Last modified April 18 06:48 EDT 2019. Contains 322209 sequences. (Running on oeis4.)