cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Clark Kimberling, Feb 03 2020

Keywords

Comments

For each partition of n, let
d = number of terms that are not repeated;
r = number of terms that are repeated.
a(n) is the number of partitions such that d > r.
Also the number of integer partitions of n with median multiplicity 1. - Gus Wiseman, Mar 20 2023

Examples

			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.

Crossrefs

For parts instead of multiplicities we have A027336
The complement is counted by A330001.
A000041 counts integer partitions, strict A000009.
A116608 counts partitions by number of distinct parts.
A237363 counts partitions with median difference 0.
Cf. A000975, A027193, A067538, A240219, A241274, A325347, A359893, A360005.

Programs

  • Mathematica
    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}]

Formula

a(n) + A241274(n) + A330001(n) = A000041(n) for n >= 0.

A330001 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, 0, 1, 1, 2, 1, 4, 2, 6, 6, 12, 13, 25, 28, 44, 54, 77, 93, 127, 155, 204, 247, 318, 390, 494, 610, 761, 937, 1172, 1442, 1783, 2194, 2693, 3292, 4028, 4917, 5946, 7221, 8700, 10490, 12584, 15106, 18004, 21523, 25537, 30399, 35945, 42635, 50219, 59382
Offset: 0

Views

Author

Clark Kimberling, Feb 03 2020

Keywords

Comments

For each partition of n, let
d = number of terms that are not repeated;
r = number of terms that are repeated.
a(n) is the number of partitions such that d < r.

Examples

			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.

Crossrefs

Cf. A000041, A241274, A329976.

Programs

  • Mathematica
    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}]

Formula

a(n) + A241274(n) + A329976(n) = A000041(n) for n >= 0.

A241518 Number of partitions p of n such that #m(1) = #m(2), where #m(i) = number of numbers in p that have multiplicity i.

Original entry on oeis.org

1, 0, 0, 1, 2, 3, 3, 4, 5, 6, 8, 12, 16, 23, 27, 41, 46, 63, 71, 96, 109, 148, 161, 219, 256, 332, 379, 500, 580, 738, 859, 1079, 1250, 1560, 1791, 2220, 2563, 3116, 3595, 4369, 5054, 6080, 7020, 8418, 9729, 11617, 13409, 15911, 18417, 21713, 25078, 29467
Offset: 0

Views

Author

Clark Kimberling, Apr 24 2014

Keywords

Examples

			a(6) counts these 3 partitions:  411, 222, 111111.

Crossrefs

Cf. A241274.

Programs

  • Mathematica
    z = 30; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]]; v[p_] :=  Length[DeleteDuplicates[Select[p, Count[p, #] == 2 &]]]; Table[Count[IntegerPartitions[n], p_ /; u[p] == v[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

Views

Author

Clark Kimberling, Feb 03 2020

Keywords

Comments

For each partition of n, let
d = number of terms that are not repeated;
r = number of terms that are repeated.
a(n) is the number of partitions such that d >= r.

Examples

			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

Crossrefs

Cf. A000041, A241274, A329976.

Programs

  • Mathematica
    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}]

Formula

a(n) + A330001(n) = A000041(n) for all n >= 0.

A330146 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, 1, 1, 3, 4, 7, 9, 13, 16, 24, 29, 39, 51, 69, 87, 118, 152, 199, 256, 330, 418, 534, 670, 838, 1046, 1296, 1603, 1960, 2412, 2936, 3588, 4342, 5288, 6364, 7713, 9272, 11186, 13389, 16117, 19213, 23032, 27408, 32715, 38810, 46176, 54582, 64692, 76286
Offset: 0

Views

Author

Clark Kimberling, Feb 03 2020

Keywords

Comments

For each partition of n, let
d = number of terms that are not repeated;
r = number of terms that are repeated.
a(n) is the number of partitions such that d <= r.

Examples

			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

Crossrefs

Cf. A000041, A241274, A329976.

Programs

  • Mathematica
    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}]

Formula

a(n) + A329976(n) = A000041(n) for all n >= 0.

A330147 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, 2, 3, 4, 4, 8, 8, 15, 20, 30, 40, 63, 78, 110, 143, 190, 238, 313, 389, 501, 621, 786, 975, 1231, 1522, 1901, 2344, 2930, 3595, 4451, 5448, 6700, 8147, 9974, 12087, 14651, 17672, 21326, 25558, 30709, 36657, 43770, 52069, 61902, 73357, 86921, 102697
Offset: 0

Views

Author

Clark Kimberling, Feb 03 2020

Keywords

Comments

For each partition of n, let
d = number of terms that are not repeated;
r = number of terms that are repeated.
a(n) is the number of partitions such that d != r.

Examples

			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) = 8

Crossrefs

Cf. A000041, A241274, A329976.

Programs

  • Mathematica
    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}]

Formula

a(n) + A241274(n) = A000041(n) for all n >= 0.
Showing 1-6 of 6 results.

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