A362609 -id:A362609 - cp's OEIS frontend

A364159 Number of integer partitions of n - 1 containing fewer 1's than any other part.

0, 1, 1, 2, 2, 3, 4, 5, 7, 9, 11, 15, 20, 23, 32, 40, 50, 61, 82, 95, 126, 149, 188, 228, 292, 337, 430, 510, 633, 748, 933, 1083, 1348, 1579, 1925, 2262, 2761, 3197, 3893, 4544, 5458, 6354, 7634, 8835, 10577, 12261, 14546, 16864, 19990, 23043, 27226, 31428

Offset: 0

Gus Wiseman, Jul 16 2023

nonn

Also integer partitions of n with least co-mode 1. Here, 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 in {a,a,b,b,b,c,c} are {a,c}.

			The a(1) = 1 through a(8) = 7 partitions:
  (1)  (11)  (21)   (31)    (41)     (51)      (61)       (71)
             (111)  (1111)  (221)    (321)     (331)      (431)
                            (11111)  (2211)    (421)      (521)
                                     (111111)  (2221)     (3221)
                                               (1111111)  (3311)
                                                          (22211)
                                                          (11111111)

For mode instead of co-mode we have A241131, ranks A360015.
The case with only one 1 is A364062, ranks A364061.
Counts partitions ranked by A364158.
Counts positions of 1's in A364191, high A364192.
A362611 counts modes in prime factorization, triangle A362614.
A362613 counts co-modes in prime factorization, triangle A362615.
Ranking and counting partitions:
- A356862 = unique mode, counted by A362608
- A359178 = unique co-mode, counted by A362610
- A362605 = multiple modes, counted by A362607
- A362606 = multiple co-modes, counted by A362609
Cf. A027336, A124943, A237984, A327472, A363486, A363487.

Table[Length[Select[IntegerPartitions[n-1],Count[#,1]

A363124 Number of integer partitions of n with more than one non-mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 3, 6, 9, 19, 28, 46, 65, 98, 132, 190, 251, 348, 451, 603, 768, 1014, 1273, 1648, 2052, 2604, 3233, 4062, 4984, 6203, 7582, 9333, 11349, 13890, 16763, 20388, 24528, 29613, 35502, 42660, 50880, 60883, 72376, 86158, 102120, 121133, 143010

Offset: 0

Gus Wiseman, May 16 2023

nonn

A non-mode in a multiset is an element that appears fewer times than at least one of the others. For example, the non-modes in {a,a,b,b,b,c,d,d,d} are {a,c}.

			The a(7) = 1 through a(10) = 9 partitions:
  (3211)  (3221)   (3321)    (5221)
          (4211)   (4221)    (5311)
          (32111)  (4311)    (6211)
                   (5211)    (32221)
                   (42111)   (43111)
                   (321111)  (52111)
                             (322111)
                             (421111)
                             (3211111)

For middle parts instead of non-modes we have A238479, complement A238478.
For modes instead of non-modes we have A362607, complement A362608.
For co-modes instead of non-modes we have A362609, complement A362610.
The complement is counted by A363125.
For non-co-modes instead of non-modes we have A363128, complement A363129.
A000041 counts integer partitions.
A008284/A058398 count partitions by length/mean.
A362611 counts modes in prime factorization, triangle A362614.
A363127 counts non-modes in prime factorization, triangle A363126.
Cf. A002865, A053263, A098859, A237984, A275870, A327472, A353836, A353863, A359893, A362612.

nmsi[ms_]:=Select[Union[ms],Count[ms,#]1&]],{n,0,30}]

A363125 Number of integer partitions of n with a unique non-mode.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 8, 9, 13, 18, 24, 24, 36, 41, 45, 57, 68, 72, 87, 95, 105, 131, 136, 149, 164, 199, 203, 232, 246, 276, 298, 335, 347, 409, 399, 467, 488, 567, 569, 636, 662, 757, 767, 878, 887, 1028, 1030, 1168, 1181, 1342, 1388, 1558, 1570, 1789, 1791

Offset: 0

Gus Wiseman, May 16 2023

nonn

A non-mode in a multiset is an element that appears fewer times than at least one of the others. For example, the non-modes in {a,a,b,b,b,c,d,d,d} are {a,c}.

			The a(4) = 1 through a(9) = 13 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (5111)     (3222)
                          (4111)    (22211)    (6111)
                          (22111)   (41111)    (22221)
                          (31111)   (221111)   (32211)
                          (211111)  (311111)   (33111)
                                    (2111111)  (51111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)

For middle parts instead of non-modes we have A238478, complement A238479.
For modes instead of non-modes we have A362608, complement A362607.
For co-modes instead of non-modes we have A362610, complement A362609.
The complement is counted by A363124.
For non-co-modes instead of non-modes we have A363129, complement A363128.
A000041 counts integer partitions.
A008284/A058398 count partitions by length/mean.
A362611 counts modes in prime factorization, triangle A362614.
A363127 counts non-modes in prime factorization, triangle A363126.
Cf. A002865, A053263, A098859, A237984, A275870, A327472, A353836, A353863, A359893, A362612.

nmsi[ms_]:=Select[Union[ms],Count[ms,#]

A363128 Number of integer partitions of n with more than one non-co-mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 6, 9, 18, 25, 44, 60, 96, 122, 188, 243, 344, 442, 615, 769, 1047, 1308, 1722, 2150, 2791, 3430, 4405, 5401, 6803, 8326, 10408, 12608, 15641, 18906, 23179, 27935, 34061, 40778, 49451, 59038, 71060, 84604, 101386, 120114, 143358

Offset: 0

Gus Wiseman, May 18 2023

nonn

We define a non-co-mode in a multiset to be an element that appears more times than at least one of the others. For example, the non-co-modes in {a,a,b,b,b,c,d,d,d} are {a,b,d}.

			The a(9) = 1 through a(12) = 9 partitions:
  (32211)  (33211)   (33221)    (43311)
           (42211)   (52211)    (44211)
           (322111)  (322211)   (62211)
                     (332111)   (422211)
                     (422111)   (522111)
                     (3221111)  (3222111)
                                (3321111)
                                (4221111)
                                (32211111)

For parts instead of multiplicities we have
For middles instead of non-co-modes we have A238479, complement A238478.
For modes instead of non-co-modes we have A362607, complement A362608.
For co-modes instead of non-co-modes we have A362609, complement A362610.
For non-modes instead of non-co-modes we have A363124, complement A363125.
The complement is counted by A363129.
A000041 counts integer partitions.
A008284/A058398 count partitions by length/mean.
A362611 counts modes in prime factorization, triangle A362614.
A362613 counts co-modes in prime factorization, triangle A362615.
A363127 counts non-modes in prime factorization, triangle A363126.
A363131 counts non-co-modes in prime factorization, triangle A363130.
Cf. A002865, A053263, A098859, A237984, A275870, A327472, A353836, A353863, A359893, A362612.

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

A363129 Number of integer partitions of n with a unique non-co-mode.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 9, 12, 18, 24, 37, 43, 64, 81, 99, 129, 162, 201, 247, 303, 364, 457, 535, 653, 765, 943, 1085, 1315, 1517, 1830, 2096, 2516, 2877, 3432, 3881, 4622, 5235, 6189, 7003, 8203, 9261, 10859, 12199, 14216, 15985, 18544, 20777, 24064, 26897

Offset: 0

Gus Wiseman, May 18 2023

nonn

We define a non-co-mode in a multiset to be an element that appears more times than at least one of the others. For example, the non-co-modes in {a,a,b,b,b,c,d,d,d} are {a,b,d}.

			The a(4) = 1 through a(9) = 18 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (3221)     (3222)
                          (3211)    (4211)     (3321)
                          (4111)    (5111)     (4221)
                          (22111)   (22211)    (4311)
                          (31111)   (32111)    (5211)
                          (211111)  (41111)    (6111)
                                    (221111)   (22221)
                                    (311111)   (33111)
                                    (2111111)  (42111)
                                               (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)

For parts instead of multiplicities we have A002133.
For middles instead of non-co-modes we have A238478, complement A238479.
For modes instead of non-co-modes we have A362608, complement A362607.
For co-modes instead of non-co-modes we have A362610, complement A362609.
For non-modes instead of non-co-modes we have A363125, complement A363124.
The complement is counted by A363128.
A000041 counts integer partitions.
A008284/A058398 count partitions by length/mean.
A362611 counts modes in prime factorization, triangle A362614.
A362613 counts co-modes in prime factorization, triangle A362615.
A363127 counts non-modes in prime factorization, triangle A363126.
A363131 counts non-co-modes in prime factorization, triangle A363130.
Cf. A002865, A053263, A098859, A237984, A275870, A327472, A353836, A353863, A359893, A362612.

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

A363224 Number of integer compositions of n in which the least part appears more than once.

Original entry on oeis.org

0, 1, 1, 5, 8, 21, 44, 94, 197, 416, 857, 1766, 3621, 7392, 15032, 30493, 61708, 124646, 251359, 506203, 1018279, 2046454, 4109534, 8246985, 16540791, 33160051, 66451484, 133122753, 266612828, 533839069, 1068701695, 2139110054, 4281063708, 8566862025

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 co-mode.

			The a(1) = 0 through a(6) = 21 compositions:
  .  (11)  (111)  (22)    (113)    (33)
                  (112)   (131)    (114)
                  (121)   (311)    (141)
                  (211)   (1112)   (222)
                  (1111)  (1121)   (411)
                          (1211)   (1113)
                          (2111)   (1122)
                          (11111)  (1131)
                                   (1212)
                                   (1221)
                                   (1311)
                                   (2112)
                                   (2121)
                                   (2211)
                                   (3111)
                                   (11112)
                                   (11121)
                                   (11211)
                                   (12111)
                                   (21111)
                                   (111111)

John Tyler Rascoe, Table of n, a(n) for n = 1..1000

The complement is counted by A105039.
For partitions instead of compositions we have A117989.
Row sums of columns k > 1 of A238342.
If all parts appear more than once we have A240085, for partitions A007690.
If the least part appears exactly twice we have A241862.
For greatest instead of least we have A363262, see triangle A238341.
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.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
Cf. A002865, A008284, A097979, A243737.

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

C_x(N)={my(x='x+O('x^N), h=sum(i=1,N,(x^(2*i)*(x-1)^3)/((x^i+x-1)*(x^(i+1)+x-1)^2))); concat([0],Vec(h))}
C_x(35) \\ John Tyler Rascoe, Jul 06 2024

G.f.: Sum_{i>0} (x^(2*i) * (x-1)^3)/((x^i + x - 1)*(x^(i+1) + x - 1)^2). - John Tyler Rascoe, Jul 06 2024

A364191 Low co-mode in the multiset of prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 2, 6, 1, 2, 1, 7, 1, 8, 3, 2, 1, 9, 2, 3, 1, 2, 4, 10, 1, 11, 1, 2, 1, 3, 1, 12, 1, 2, 3, 13, 1, 14, 5, 3, 1, 15, 2, 4, 1, 2, 6, 16, 1, 3, 4, 2, 1, 17, 2, 18, 1, 4, 1, 3, 1, 19, 7, 2, 1, 20, 2, 21, 1, 2, 8, 4, 1, 22, 3, 2, 1

Offset: 1

Gus Wiseman, Jul 16 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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 in {a,a,b,b,b,c,c} are {a,c}.
Extending the terminology of A124943, the "low co-mode" in a multiset is the least co-mode.

			The prime indices of 2100 are {1,1,2,3,3,4}, with co-modes {2,4}, so a(2100) = 2.

For prime factors instead of indices we have A067695, high A359612.
For mode instead of co-mode we have A363486, high A363487, triangle A363952.
For median instead of co-mode we have A363941, high A363942.
Positions of 1's are A364158, counted by A364159.
The high version is A364192 = positions of 1's in A364061.
A112798 lists prime indices, length A001222, sum A056239.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
Ranking and counting partitions:
- A356862 = unique mode, counted by A362608
- A359178 = unique co-mode, counted by A362610
- A362605 = multiple modes, counted by A362607
- A362606 = multiple co-modes, counted by A362609
Cf. A124943, A241131, A327473, A327476, A360005, A360015, A363488.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==1,0,Min[comodes[prix[n]]]],{n,30}]

a(n) = A000720(A067695(n)).

A067695(n) = A000040(a(n)).

A364192 High (i.e., greatest) co-mode in the multiset of prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 3, 1, 7, 1, 8, 3, 4, 5, 9, 2, 3, 6, 2, 4, 10, 3, 11, 1, 5, 7, 4, 2, 12, 8, 6, 3, 13, 4, 14, 5, 3, 9, 15, 2, 4, 1, 7, 6, 16, 1, 5, 4, 8, 10, 17, 3, 18, 11, 4, 1, 6, 5, 19, 7, 9, 4, 20, 2, 21, 12, 2, 8, 5, 6, 22, 3, 2

Offset: 1

Gus Wiseman, Jul 16 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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 in {a,a,b,b,b,c,c} are {a,c}.
Extending the terminology of A124943, the "high co-mode" in a multiset is the greatest co-mode.

			The prime indices of 2100 are {1,1,2,3,3,4}, with co-modes {2,4}, so a(2100) = 4.

For prime factors instead of indices we have A359612, low A067695.
For mode instead of co-mode we have A363487 (triangle A363953), low A363486 (triangle A363952).
The version for median instead of co-mode is A363942, low A363941.
Positions of 1's are A364061, counted by A364062.
The low version is A364191, 1's at A364158 (counted by A364159).
A112798 lists prime indices, length A001222, sum A056239.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
Ranking and counting partitions:
- A356862 = unique mode, counted by A362608
- A359178 = unique co-mode, counted by A362610
- A362605 = multiple modes, counted by A362607
- A362606 = multiple co-modes, counted by A362609
Cf. A241131, A327473, A327476, A360005, A360015, A362612, A363740.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==1,0,Max[comodes[prix[n]]]],{n,30}]

a(n) = A000720(A359612(n)).

A359612(n) = A000040(a(n)).

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}]

cp's OEIS Frontend

A364159 Number of integer partitions of n - 1 containing fewer 1's than any other part.

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A363124 Number of integer partitions of n with more than one non-mode.

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A363125 Number of integer partitions of n with a unique non-mode.

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A363128 Number of integer partitions of n with more than one non-co-mode.

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A363129 Number of integer partitions of n with a unique non-co-mode.

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A363224 Number of integer compositions of n in which the least part appears more than once.

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A364191 Low co-mode in the multiset of prime indices of n.

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A364192 High (i.e., greatest) co-mode in the multiset of prime indices of n.

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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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