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A228125 Triangle read by rows: T(n,k) = number of semistandard Young tableaux with sum of entries equal to n and shape of tableau a partition of k. 5
1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 1, 1, 5, 7, 5, 2, 1, 1, 6, 10, 9, 5, 2, 1, 1, 7, 14, 16, 10, 5, 2, 1, 1, 8, 19, 24, 19, 11, 5, 2, 1, 1, 9, 24, 37, 32, 21, 11, 5, 2, 1, 1, 10, 30, 51, 52, 38, 22, 11, 5, 2, 1, 1, 11, 37, 71, 79, 66, 41, 23, 11, 5, 2, 1, 1, 12, 44, 93, 117, 106, 74, 43, 23, 11, 5, 2, 1, 1, 13, 52, 122, 166, 166, 125, 80, 44, 23, 11, 5, 2, 1, 1, 14, 61, 153, 231, 251, 204, 139, 83, 45, 23, 11, 5, 2, 1, 1, 15, 70, 193, 311, 367, 322, 236, 147, 85, 45, 23, 11, 5, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Row sums equal A003293.

Reverse of rows seem to converge to A005986: 1, 2, 5, 11, 23, 45, 87, 160, ...

LINKS

Table of n, a(n) for n=1..136.

EXAMPLE

T(6,3) = 7 since the 7 SSYT with sum of entries = 6 and shape any partition of 3 are

114 , 123 , 222 , 11 ,  12  , 13 ,   1

                  4     3     2      2

                                     3

Triangle starts:

1;

1,  1;

1,  2,  1;

1,  3,  2,  1;

1,  4,  4,  2,  1;

1,  5,  7,  5,  2,  1;

1,  6, 10,  9,  5,  2,  1;

1,  7, 14, 16, 10,  5,  2,  1;

1,  8, 19, 24, 19, 11,  5,  2, 1;

1,  9, 24, 37, 32, 21, 11,  5, 2, 1;

1, 10, 30, 51, 52, 38, 22, 11, 5, 2, 1;

MATHEMATICA

hooklength[(par_)?PartitionQ]:=Table[Count[par, q_ /; q>=j] +1-i +par[[i]] -j, {i, Length[par]}, {j, par[[i]]} ];

Table[Tr[(SeriesCoefficient[q^(#1 . Range[Length[#1]])/Times @@ (1-q^#1&) /@ Flatten[hooklength[#1]], {q, 0, w}]&) /@ Partitions[n]], {w, 24}, {n, w}]

CROSSREFS

Cf. A003293, A005986.

Sequence in context: A092905 A052509 A172119 * A227588 A093628 A186807

Adjacent sequences:  A228122 A228123 A228124 * A228126 A228127 A228128

KEYWORD

nonn,tabl

AUTHOR

Wouter Meeussen, Aug 11 2013

STATUS

approved

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Last modified June 25 17:47 EDT 2019. Contains 324353 sequences. (Running on oeis4.)