A175595 Square array A(n,t), n>=0, t>=0, read by antidiagonals: A(n,t) is the number of t-core partitions of n.

1, 1, 1, 1, 0, 2, 1, 1, 0, 3, 1, 1, 0, 0, 5, 1, 1, 2, 1, 0, 7, 1, 1, 2, 0, 0, 0, 11, 1, 1, 2, 3, 2, 0, 0, 15, 1, 1, 2, 3, 1, 1, 1, 0, 22, 1, 1, 2, 3, 5, 3, 2, 0, 0, 30, 1, 1, 2, 3, 5, 2, 3, 0, 0, 0, 42, 1, 1, 2, 3, 5, 7, 6, 3, 1, 0, 0, 56, 1, 1, 2, 3, 5, 7, 5, 5, 4, 2, 1, 0, 77, 1, 1, 2, 3, 5, 7, 11, 9, 7, 4, 2, 0, 0, 101

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

Columns t=0-12 give A000041, A000007, A010054, A033687, A045831, A053723, A081622, A053724, A182803, A182804, A182805, A053691, A192061.

Rows n=0-1 give A000012, A060576.

Diagonal gives A000094(n+1) for n>0.

Upper diagonal gives A000041.

Lower diagonal (conjectured) gives A086642 for n>0.

Sequence in context: A143810 A128589 A130162 * A175417 A136481 A100218

Adjacent sequences: A175592 A175593 A175594 * A175596 A175597 A175598

Keyword

nonn,tabl

Author

Alois P. Heinz, Dec 03 2010

Extensions

Additional references from N. J. A. Sloane, Jun 21 2011

Status

approved