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%I #42 Mar 20 2019 04:23:48
%S 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,
%T 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,
%U 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
%N Square array A(n,t), n>=0, t>=0, read by antidiagonals: A(n,t) is the number of t-core partitions of n.
%C 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.
%D 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.
%D James, Gordon; and Kerber, Adalbert, The Representation Theory of the Symmetric Group. Addison-Wesley Publishing Co., Reading, Mass., 1981.
%H Alois P. Heinz, <a href="/A175595/b175595.txt">Antidiagonals n = 0..140, flattened</a>
%H G. E. Andrews and F. Garvan, <a href="https://doi.org/10.1090/S0273-0979-1988-15637-6">Dyson's crank of a partition</a>, Bull. Amer. Math. Soc., 18 (1988), 167-171.
%H A. O. L. Atkins and F. G. Garvan, <a href="https://arxiv.org/abs/math/0208050">Relations between the ranks and cranks of partitions</a>, arXiv:math/0208050 [math.NT], 2002.
%H A. O. L. Atkins and F. G. Garvan, <a href="https://doi.org/10.1023/A:1026219901284">Relations between the ranks and cranks of partitions</a>, Rankin memorial issues. Ramanujan J. 7 (2003), 343-366.
%H Shichao Chen, <a href="https://doi.org/10.1007/s11139-007-9045-5">Arithmetical properties of the number of t-core partitions</a>, The Ramanujan Journal, 18 (2007), no. 1, 103-112, DOI: 10.1007/s11139-007-9045-5.
%H F. G. Garvan, <a href="https://doi.org/10.1090/S0002-9947-1990-1012520-8">The crank of partitions mod 8, 9 and 10</a>, Trans. Amer. Math. Soc. 322 (1990), 79-94.
%H F. G. Garvan, <a href="https://doi.org/10.1112/plms/s3-66.3.449">Some congruences for partitions that are p-cores</a>, Proc. London Math. Soc. 66 (1993), 449-478.
%H F. G. Garvan, <a href="https://doi.org/10.1017/S0004972700019481">More cranks and t-cores</a>, Bull. Austral. Math. Soc. 63 (2001), 379-391.
%H F. G. Garvan, D. Kim and D. Stanton, <a href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN00210752X">Cranks and t-cores</a>, Inventiones Math. 101 (1990) 1-17.
%H Andrew Granville and Ken Ono, <a href="https://doi.org/10.1090/S0002-9947-96-01481-X">Defect Zero p-blocks for Finite Simple Groups</a>, Transactions of the American Mathematical Society, Vol. 348 (1996), pp. 331-347.
%H Ben Kane, <a href="http://www.math.ru.nl/~bkane/papers/partition/nottcore.pdf">Sums of Triangular Numbers and t-Core Partitions</a>, Journal of Combinatorics and Number Theory, 1 (2009), no.1, 59-64.
%H B. Kim, <a href="https://doi.org/10.1016/j.disc.2009.09.024">On inequalities and linear relations for 7-core partitions</a>, Discrete Math., 310 (2010), 861-868.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%F G.f. of column t: Product_{i>=1} (1-x^(t*i))^t/(1-x^i).
%F Column t is the Euler transform of period t sequence [1, .., 1, 1-t, ..].
%e A(4,3) = 2, because there are 2 partitions of 4 such that no hook number is a multiple of 3:
%e (1) 2 | 4 1
%e +1 | 2
%e +1 | 1
%e -------+-----
%e (2) 3 | 4 2 1
%e +1 | 1
%e Square array A(n,t) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 0, 1, 1, 1, 1, 1, 1, ...
%e 2, 0, 0, 2, 2, 2, 2, 2, ...
%e 3, 0, 1, 0, 3, 3, 3, 3, ...
%e 5, 0, 0, 2, 1, 5, 5, 5, ...
%e 7, 0, 0, 1, 3, 2, 7, 7, ...
%e 11, 0, 1, 2, 3, 6, 5, 11, ...
%e 15, 0, 0, 0, 3, 5, 9, 8, ...
%p with(numtheory):
%p A:= proc(n, t) option remember; `if`(n=0, 1,
%p add(add(`if`(t=0 or irem(d, t)=0, d-d*t, d),
%p d=divisors(j))*A(n-j, t), j=1..n)/n)
%p end:
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%p (From _N. J. A. Sloane_, Jun 21, 2011: to get M terms of the series for t-core partitions:)
%p M:=60;
%p f:=proc(t) global M; local q,i,t1;
%p t1:=1;
%p for i from 1 to M+1 do
%p t1:=series(t1*(1-q^(i*t))^t,q,M);
%p t1:=series(t1/(1-q^i),q,M);
%p od;
%p t1;
%p end;
%p # then for example seriestolist(f(5));
%t n = 13; f[t_] = (1-x^(t*k))^t/(1-x^k); f[0] = 1/(1-x^k);
%t 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. *)
%Y Columns t=0-12 give A000041, A000007, A010054, A033687, A045831, A053723, A081622, A053724, A182803, A182804, A182805, A053691, A192061.
%Y Rows n=0-1 give A000012, A060576.
%Y Diagonal gives A000094(n+1) for n>0.
%Y Upper diagonal gives A000041.
%Y Lower diagonal (conjectured) gives A086642 for n>0.
%K nonn,tabl
%O 0,6
%A _Alois P. Heinz_, Dec 03 2010
%E Additional references from _N. J. A. Sloane_, Jun 21 2011
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