A329976
Number of partitions p of n such that (number of numbers in p that have multiplicity 1) > (number of numbers in p having multiplicity > 1).
Original entry on oeis.org
0, 1, 1, 2, 2, 3, 4, 6, 9, 14, 18, 27, 38, 50, 66, 89, 113, 145, 186, 234, 297, 374, 468, 585, 737, 912, 1140, 1407, 1758, 2153, 2668, 3254, 4007, 4855, 5946, 7170, 8705, 10451, 12626, 15068, 18125, 21551, 25766, 30546, 36365, 42958, 50976, 60062, 70987
Offset: 0
The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
These have d > r: 6, 51, 42, 321
These have d = r: 411, 3222, 21111
These have d < r: 33, 222, 2211, 111111
Thus, a(6) = 4.
For parts instead of multiplicities we have
A027336The complement is counted by
A330001.
A116608 counts partitions by number of distinct parts.
A237363 counts partitions with median difference 0.
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z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[Count[IntegerPartitions[n], p_ /; d[p] > r[p]], {n, 0, z}]
A241274
Number of partitions p of n such that (number of numbers in p that have multiplicity 1) = (number of numbers in p having multiplicity > 1).
Original entry on oeis.org
1, 0, 0, 0, 1, 3, 3, 7, 7, 10, 12, 16, 14, 23, 25, 33, 41, 59, 72, 101, 126, 171, 216, 280, 344, 436, 535, 666, 788, 970, 1153, 1394, 1649, 1996, 2336, 2796, 3326, 3965, 4689, 5627, 6629, 7926, 9404, 11192, 13273, 15777, 18637, 22057, 26067, 30672, 36122
Offset: 0
a(6) counts these 3 partitions: 411, 3111, 21111.
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z = 30; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]]; e[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[Count[IntegerPartitions[n], p_ /; u[p] == e[p]], {n, 0, z}]
A330145
Number of partitions p of n such that (number of numbers in p that have multiplicity 1) >= (number of numbers in p having multiplicity > 1).
Original entry on oeis.org
1, 1, 1, 2, 3, 6, 7, 13, 16, 24, 30, 43, 52, 73, 91, 122, 154, 204, 258, 335, 423, 545, 684, 865, 1081, 1348, 1675, 2073, 2546, 3123, 3821, 4648, 5656, 6851, 8282, 9966, 12031, 14416, 17315, 20695, 24754, 29477, 35170, 41738, 49638, 58735, 69613, 82119
Offset: 0
The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
These have d > r: 6, 51, 42, 321
These have d = r: 411, 3222, 21111
These have d < r: 33, 222, 2211, 111111
Thus, a(6) = 7
-
z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[Count[IntegerPartitions[n], p_ /; d[p] >= r[p]], {n, 0, z}]
Showing 1-3 of 3 results.
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