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
Examples
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, ...
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.
Crossrefs
Programs
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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. *)
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, ..].
Extensions
Additional references from N. J. A. Sloane, Jun 21 2011
Comments