OFFSET
0,6
COMMENTS
A partition of n is a t-core partition if none of the hook numbers associated to the Ferrers-Young diagram is a multiple of t. See Chen reference for definitions.
REFERENCES
Garvan, F. G., A number-theoretic crank associated with open bosonic strings. In Number Theory and Cryptography (Sydney, 1989), 221-226, London Math. Soc. Lecture Note Ser., 154, Cambridge Univ. Press, Cambridge, 1990.
James, Gordon; and Kerber, Adalbert, The Representation Theory of the Symmetric Group. Addison-Wesley Publishing Co., Reading, Mass., 1981.
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.
A. O. L. Atkins and F. G. Garvan, Relations between the ranks and cranks of partitions, arXiv:math/0208050 [math.NT], 2002.
A. O. L. Atkins and F. G. Garvan, Relations between the ranks and cranks of partitions, Rankin memorial issues. Ramanujan J. 7 (2003), 343-366.
Shichao Chen, Arithmetical properties of the number of t-core partitions, The Ramanujan Journal, 18 (2007), no. 1, 103-112, DOI: 10.1007/s11139-007-9045-5.
F. G. Garvan, The crank of partitions mod 8, 9 and 10, Trans. Amer. Math. Soc. 322 (1990), 79-94.
F. G. Garvan, Some congruences for partitions that are p-cores, Proc. London Math. Soc. 66 (1993), 449-478.
F. G. Garvan, More cranks and t-cores, Bull. Austral. Math. Soc. 63 (2001), 379-391.
F. G. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.
Andrew Granville and Ken Ono, Defect Zero p-blocks for Finite Simple Groups, Transactions of the American Mathematical Society, Vol. 348 (1996), pp. 331-347.
Ben Kane, Sums of Triangular Numbers and t-Core Partitions, Journal of Combinatorics and Number Theory, 1 (2009), no.1, 59-64.
B. Kim, On inequalities and linear relations for 7-core partitions, Discrete Math., 310 (2010), 861-868.
N. J. A. Sloane, Transforms.
FORMULA
G.f. of column t: Product_{i>=1} (1-x^(t*i))^t/(1-x^i).
Column t is the Euler transform of period t sequence [1, .., 1, 1-t, ..].
EXAMPLE
A(4,3) = 2, because there are 2 partitions of 4 such that no hook number is a multiple of 3:
(1) 2 | 4 1
+1 | 2
+1 | 1
-------+-----
(2) 3 | 4 2 1
+1 | 1
Square array A(n,t) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 0, 1, 1, 1, 1, 1, 1, ...
2, 0, 0, 2, 2, 2, 2, 2, ...
3, 0, 1, 0, 3, 3, 3, 3, ...
5, 0, 0, 2, 1, 5, 5, 5, ...
7, 0, 0, 1, 3, 2, 7, 7, ...
11, 0, 1, 2, 3, 6, 5, 11, ...
15, 0, 0, 0, 3, 5, 9, 8, ...
MAPLE
with(numtheory):
A:= proc(n, t) option remember; `if`(n=0, 1,
add(add(`if`(t=0 or irem(d, t)=0, d-d*t, d),
d=divisors(j))*A(n-j, t), j=1..n)/n)
end:
seq(seq(A(n, d-n), n=0..d), d=0..14);
(From N. J. A. Sloane, Jun 21, 2011: to get M terms of the series for t-core partitions:)
M:=60;
f:=proc(t) global M; local q, i, t1;
t1:=1;
for i from 1 to M+1 do
t1:=series(t1*(1-q^(i*t))^t, q, M);
t1:=series(t1/(1-q^i), q, M);
od;
t1;
end;
# then for example seriestolist(f(5));
MATHEMATICA
n = 13; f[t_] = (1-x^(t*k))^t/(1-x^k); f[0] = 1/(1-x^k);
s[t_] := CoefficientList[ Series[ Product[ f[t], {k, 1, n}], {x, 0, n}], x]; m = Table[ PadRight[ s[t], n+1], {t, 0, n}]; Flatten[ Table[ m[[j+1-k, k]], {j, n+1}, {k, j}]] (* Jean-François Alcover, Jul 25 2011, after g.f. *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Dec 03 2010
EXTENSIONS
Additional references from N. J. A. Sloane, Jun 21 2011
STATUS
approved