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Number of partitions satisfying cn(0,5) + cn(2,5) <= cn(1,5) and cn(0,5) + cn(2,5) <= cn(4,5) and cn(0,5) + cn(3,5) <= cn(1,5) and cn(0,5) + cn(3,5) <= cn(4,5).
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%I #12 Oct 10 2024 15:16:39

%S 1,1,1,1,2,2,3,4,6,7,10,12,18,22,27,37,45,58,74,89,118,140,177,215,

%T 265,333,397,487,592,711,881,1041,1263,1507,1803,2185,2578,3080,3656,

%U 4321,5185,6062,7193,8450,9931,11772,13706,16105,18827,21945,25817,29880

%N Number of partitions satisfying cn(0,5) + cn(2,5) <= cn(1,5) and cn(0,5) + cn(2,5) <= cn(4,5) and cn(0,5) + cn(3,5) <= cn(1,5) and cn(0,5) + cn(3,5) <= cn(4,5).

%C For a given partition cn(i,n) means the number of its parts equal to i modulo n.

%C Short: 0 + 2 <= 1 and 0 + 2 <= 4 and 0 + 3 <= 1 and 0 + 3 <= 4 (ZBBAA).

%t okQ[p_] := Module[{c},

%t c[k_] := c[k] = Count[Mod[p, 5], k];

%t c[0] + c[2] <= c[1] &&

%t c[0] + c[2] <= c[4] &&

%t c[0] + c[3] <= c[1] &&

%t c[0] + c[3] <= c[4]];

%t a[n_] := a[n] = Count[okQ /@ IntegerPartitions[n], True];

%t Table[Print[n, " ", a[n]]; a[n], {n, 1, 45}] (* _Jean-François Alcover_, Oct 10 2024 *)

%K nonn

%O 0,5

%A _Olivier Gérard_

%E a(0)=1 prepended by _Alois P. Heinz_, Oct 10 2024