A129130 Number of triples of standard tableaux with the same shape of height less than or equal to three.

1, 1, 2, 10, 63, 531, 6201, 70477, 897149, 12772405, 188334604, 2939523104, 47902337803, 809518276503, 14134544880444, 252955559204532, 4651455689358657, 87356706437180529, 1669767921758484702, 32535861588779772366, 643610071378067450895, 12905615915176151132595

Offset

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.

Simone Severini and Ferenc Szollosi, A further look into combinatorial orthogonality, arXiv:0709.3651 [math.CO], 2007-2008. Mentions this sequence, see p. 7.

Wikipedia, Hook length formula.

Formula

sum_lambda f_lambda^3 where the sum is over lambda partitions of length less than or equal to 3 and f_lambda is the number of standard tableaux of shape lambda

Example

f_111 = f_3 = 1, f_21 = 2 therefore a(3) = f_111^3 + f_21^3 + f_3^3 = 1+8+1 = 10

Prog

(PARI) \\ HLF is hook length formula of partition.
HLF(p) = { my(v=vector(if(#p, p[#p], 0)), r=1); p=vecsort(p); for(i=1, #p, my(t=p[i]); for(j=1, t, v[j]++; r *= v[j] + t - j )); vecsum(v)!/r }
a(n) = { my(s=0); forpart(p=n, s+=HLF(p)^3, , 3); s } \\ Andrew Howroyd, Jul 05 2026

Crossrefs

Cf. A001006, A005802, A003161.

Sequence in context: A378466 A175962 A183165 * A245519 A303483 A186268

Adjacent sequences: A129127 A129128 A129129 * A129131 A129132 A129133

Keyword

nonn,changed

Author

Mike Zabrocki, Mar 30 2007

Extensions

a(19) onward from Andrew Howroyd, Jul 05 2026

Status

approved