A363224 -id:A363224 - cp's OEIS frontend

A363262 Number of integer compositions of n in which the greatest part appears more than once.

0, 1, 1, 2, 4, 9, 18, 37, 73, 145, 287, 570, 1134, 2264, 4526, 9061, 18152, 36374, 72884, 146011, 292416, 585422, 1171632, 2344136, 4688821, 9376832, 18749169, 37485358, 74939850, 149813328, 299492966, 598729533, 1196987066, 2393137399, 4784846896, 9567357951

Offset: 1

Gus Wiseman, Jun 04 2023

nonn

Also the number of multisets of length n covering an initial interval of positive integers with more than one mode.

			The a(2) = 1 through a(6) = 9 compositions:
  (11)  (111)  (22)    (122)    (33)
               (1111)  (212)    (222)
                       (221)    (1122)
                       (11111)  (1212)
                                (1221)
                                (2112)
                                (2121)
                                (2211)
                                (111111)

For partitions instead of compositions we have A002865.
The complement is counted by A097979 shifted left.
Row sums of columns k > 1 of A238341.
If all parts appear more than once we have A240085, for partitions A007690.
If the greatest part appears exactly twice we have A243737.
For least instead of greatest we have A363224, see triangle A238342.
A000041 counts integer partitions, strict A000009.
A032020 counts strict compositions.
A067029 gives last exponent in prime factorization, first A071178.
A261982 counts compositions with some part appearing more than once.
Cf. A008284, A105039, A117989, A362607, A362608, A362612, A362614.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Count[#,Max@@#]>1&]],{n,15}]

A363132 Number of integer partitions of 2n such that 2*(minimum) = (mean).

Original entry on oeis.org

0, 0, 1, 2, 5, 6, 15, 14, 32, 34, 65, 55, 150, 100, 225, 237, 425, 296, 824, 489, 1267, 1133, 1809, 1254, 4018, 2142, 4499, 4550, 7939, 4564, 14571, 6841, 18285, 16047, 23408, 17495, 52545, 21636, 49943, 51182, 92516, 44582, 144872, 63260, 175318, 169232, 205353

Offset: 0

Gus Wiseman, May 23 2023

nonn

Equivalently, n = (length)*(minimum).

			The a(2) = 1 through a(7) = 14 partitions:
  (31)  (321)  (62)    (32221)  (93)      (3222221)
        (411)  (3221)  (33211)  (552)     (3322211)
               (3311)  (42211)  (642)     (3332111)
               (4211)  (43111)  (732)     (4222211)
               (5111)  (52111)  (822)     (4322111)
                       (61111)  (322221)  (4331111)
                                (332211)  (4421111)
                                (333111)  (5222111)
                                (422211)  (5321111)
                                (432111)  (5411111)
                                (441111)  (6221111)
                                (522111)  (6311111)
                                (531111)  (7211111)
                                (621111)  (8111111)
                                (711111)

Removing the factor 2 gives A099777.
Taking maximum instead of mean and including odd indices gives A118096.
For length instead of mean and including odd indices we have A237757.
For (maximum) = 2*(mean) see A361851, A361852, A361853, A361854, A361855.
For median instead of mean we have A361861.
These partitions have ranks A363133.
For maximum instead of minimum we have A363218.
For median instead of minimum we have A363224.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
A268192 counts partitions by complement size, ranks A326844.
Cf. A053263, A111907, A237753, A237755, A237824, A327482, A349156, A361906, A361907, A363134.

Table[Length[Select[IntegerPartitions[2n],2*Min@@#==Mean[#]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
def A363132(n): return sum(1 for s,p in partitions(n<<1,m=n,size=True) if n==s*min(p,default=0)) if n else 0 # Chai Wah Wu, Sep 21 2023

a(31)-a(46) from Chai Wah Wu, Sep 21 2023

A363263 Number of integer partitions of n covering an initial interval of positive integers with a unique co-mode.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 2, 4, 4, 5, 7, 10, 8, 13, 13, 15, 19, 25, 24, 35, 35, 43, 50, 61, 59, 79, 83, 98, 111, 137, 137, 176, 187, 219, 240, 284, 298, 360, 385, 444, 485, 568, 600, 706, 763, 867, 951, 1088, 1168, 1345, 1453, 1641, 1792, 2023, 2179, 2467, 2673, 2988

Offset: 0

Gus Wiseman, Jun 06 2023

nonn

We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

			The a(1) = 1 through a(10) = 7 partitions:
  1  11  111  211   221    21111   2221     22211     22221      33211
              1111  2111   111111  22111    221111    32211      222211
                    11111          211111   2111111   2211111    322111
                                   1111111  11111111  21111111   2221111
                                                      111111111  22111111
                                                                 211111111
                                                                 1111111111
The a(9) = 5 through a(12) = 8 partitions:
  (22221)      (33211)       (33221)        (2222211)
  (32211)      (222211)      (222221)       (3222111)
  (2211111)    (322111)      (322211)       (3321111)
  (21111111)   (2221111)     (332111)       (32211111)
  (111111111)  (22111111)    (2222111)      (222111111)
               (211111111)   (3221111)      (2211111111)
               (1111111111)  (22211111)     (21111111111)
                             (221111111)    (111111111111)
                             (2111111111)
                             (11111111111)

For parts instead of multiplicities we have A087897, complement A000009.
For multisets instead of partitions we have A105039, complement A363224.
The complement is counted by A363264.
For mode we have A363484, complement A363485.
A000041 counts integer partitions, A000009 covering an initial interval.
A097979 counts normal multisets with a unique mode, complement A363262.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A025147, A096765, A117989, A243737, A362612.

comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==0,0,Length[Select[IntegerPartitions[n],Union[#]==Range[Max@@#]&&Length[comsi[#]]==1&]]],{n,0,30}]

A363264 Number of integer partitions of n covering an initial interval of positive integers with a more than one co-mode.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 1, 2, 3, 3, 2, 7, 5, 9, 12, 13, 13, 22, 19, 29, 33, 39, 43, 63, 63, 82, 94, 111, 119, 159, 164, 203, 229, 272, 301, 370, 400, 479, 538, 628, 692, 826, 904, 1053, 1181, 1353, 1502, 1742, 1919, 2205, 2456, 2790, 3097, 3539, 3911, 4435, 4929

Offset: 0

Gus Wiseman, Jun 06 2023

nonn

We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

For parts instead of multiplicities we have A000009, complement A087897.
For multisets instead of partitions we have A363224, complement A105039.
The complement is counted by A363263.
For mode we have A363485, complement A363484.
A000041 counts integer partitions, A000009 covering an initial interval.
A067029 counts minima in prime factorization, co-modes A362613.
A071178 counts maxima in prime factorization, modes A362611.
A097979 counts normal multisets with a unique mode, complement A363262.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A096765, A117989, A243737, A275870, A362612.

comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==0,0,Length[Select[IntegerPartitions[n],Union[#]==Range[Max@@#]&&Length[comsi[#]]>1&]]],{n,0,30}]

A363219 Twice the median of the conjugate of the integer partition with Heinz number n.

Original entry on oeis.org

0, 2, 2, 4, 2, 3, 2, 6, 4, 2, 2, 4, 2, 2, 4, 8, 2, 5, 2, 2, 3, 2, 2, 5, 4, 2, 6, 2, 2, 4, 2, 10, 2, 2, 4, 6, 2, 2, 2, 2, 2, 3, 2, 2, 6, 2, 2, 6, 4, 4, 2, 2, 2, 7, 4, 2, 2, 2, 2, 4, 2, 2, 4, 12, 3, 2, 2, 2, 2, 4, 2, 7, 2, 2, 6, 2, 4, 2, 2, 2, 8, 2, 2, 3, 2, 2

Offset: 1

Gus Wiseman, May 25 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The partition (4,2,1) has Heinz number 42 and conjugate (3,2,1,1) with median 3/2, so a(42) = 3.

Twice the row media of A321649 or A321650.
For mean instead of twice median we have A326839/A326840.
This is the conjugate version of A360005.
A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 is partition conjugation in terms of Heinz numbers.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A352491 gives n minus Heinz number of conjugate.
A363220 counts partitions with same median as conjugate.
Cf. A046682, A321648, A325040, A326567/A326568, A326848, A362617, A362618, A363223, A363224.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[If[n==1,0,2*Median[conj[prix[n]]]],{n,100}]

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A363262 Number of integer compositions of n in which the greatest part appears more than once.

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A363132 Number of integer partitions of 2n such that 2*(minimum) = (mean).

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A363263 Number of integer partitions of n covering an initial interval of positive integers with a unique co-mode.

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A363264 Number of integer partitions of n covering an initial interval of positive integers with a more than one co-mode.

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A363219 Twice the median of the conjugate of the integer partition with Heinz number n.

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