A350837 -id:A350837 - cp's OEIS frontend

A350842 Number of integer partitions of n with no difference -2.

1, 1, 2, 3, 4, 6, 9, 12, 16, 24, 30, 40, 54, 69, 89, 118, 146, 187, 239, 297, 372, 468, 575, 711, 880, 1075, 1314, 1610, 1947, 2359, 2864, 3438, 4135, 4973, 5936, 7090, 8466, 10044, 11922, 14144, 16698, 19704, 23249, 27306, 32071, 37639, 44019, 51457, 60113

Offset: 0

Gus Wiseman, Jan 20 2022

nonn

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (211)   (41)     (51)      (52)
                    (1111)  (221)    (222)     (61)
                            (2111)   (321)     (322)
                            (11111)  (411)     (511)
                                     (2211)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (22111)
                                               (211111)
                                               (1111111)

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

Heinz number rankings are in parentheses below.
The version for no difference 0 is A000009.
The version for subsets of prescribed maximum is A005314.
The version for all differences < -2 is A025157, non-strict A116932.
The version for all differences > -2 is A034296, strict A001227.
The opposite version is A072670.
The version for no difference -1 is A116931 (A319630), strict A003114.
The multiplicative version is A350837 (A350838), strict A350840.
The strict case is A350844.
The complement for quotients is counted by A350846 (A350845).
A000041 = integer partitions.
A027187 = partitions of even length.
A027193 = partitions of odd length (A026424).
A323092 = double-free partitions (A320340), strict A120641.
A325534 = separable partitions (A335433).
A325535 = inseparable partitions (A335448).
A350839 = partitions with a gap and conjugate gap (A350841).
Cf. A000070, A000929, A001511, A003242, A007359, A018819, A040039, A045690, A045691, A101417, A154402, A323093.

Table[Length[Select[IntegerPartitions[n],FreeQ[Differences[#],-2]&]],{n,0,30}]

A351003 Number of integer partitions y of n such that y_i = y_{i+1} for all even i.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 28, 36, 42, 51, 62, 75, 88, 106, 124, 147, 173, 202, 236, 278, 320, 371, 431, 497, 572, 661, 756, 867, 993, 1132, 1291, 1474, 1672, 1898, 2155, 2439, 2756, 3117, 3512, 3957, 4458, 5008, 5624, 6316, 7072, 7919, 8862, 9899

Offset: 0

Gus Wiseman, Jan 31 2022

nonn

			The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (311)    (51)      (61)
                    (1111)  (2111)   (222)     (322)
                            (11111)  (411)     (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)

The ordered version (compositions) is A027383.
The version for unequal instead of equal is A122135, even-length A351008.
For odd instead of even indices we have A351004, even-length A035363.
Requiring inequalities at odd positions gives A351006, even-length A351007.
The even-length case is A351012.
Cf. A000070, A018819, A088218, A101417, A122129, A350837, A351005.

Table[Length[Select[IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,10}]

A350839 Number of integer partitions of n with a difference < -1 and a conjugate difference < -1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 3, 7, 11, 17, 26, 39, 54, 81, 108, 148, 201, 269, 353, 467, 601, 779, 995, 1272, 1605, 2029, 2538, 3171, 3941, 4881, 6012, 7405, 9058, 11077, 13478, 16373, 19817, 23953, 28850, 34692, 41599, 49802, 59461, 70905, 84321, 100155, 118694

Offset: 0

Gus Wiseman, Jan 24 2022

nonn

We define a difference of a partition to be a difference of two adjacent parts.

			The a(5) = 1 through a(10) = 17 partitions:
  (311)  (411)   (511)    (422)     (522)      (622)
         (3111)  (4111)   (611)     (711)      (811)
                 (31111)  (3311)    (4221)     (4222)
                          (4211)    (4311)     (4411)
                          (5111)    (5211)     (5221)
                          (41111)   (6111)     (5311)
                          (311111)  (33111)    (6211)
                                    (42111)    (7111)
                                    (51111)    (42211)
                                    (411111)   (43111)
                                    (3111111)  (52111)
                                               (61111)
                                               (331111)
                                               (421111)
                                               (511111)
                                               (4111111)
                                               (31111111)

Allowing -1 gives A144300 = non-constant partitions.
Taking one of the two conditions gives A239955, ranked by A073492, A065201.
These partitions are ranked by A350841.
A000041 = integer partitions, strict A000009.
A034296 = flat (contiguous) partitions, strict A001227.
A073491 = numbers whose prime indices have no gaps, strict A137793.
A090858 = partitions with a single hole, ranked by A325284.
A116931 = partitions with differences != -1, strict A003114.
A116932 = partitions with differences != -1 or -2, strict A025157.
A277103 = partitions with the same number of odd parts as their conjugate.
A350837 = partitions with no adjacent doublings, strict A350840.
A350842 = partitions with differences != -2, strict A350844, sets A005314.
Cf. A000070, A000929, A008284, A040039, A183558, A319630, A321440, A350838.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],(Min@@Differences[#]<-1)&&(Min@@Differences[conj[#]]<-1)&]],{n,0,30}]

A350844 Number of strict integer partitions of n with no difference -2.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 4, 4, 7, 7, 8, 11, 12, 15, 18, 21, 23, 31, 32, 40, 45, 54, 59, 73, 78, 94, 106, 122, 136, 161, 177, 203, 231, 259, 293, 334, 372, 417, 476, 525, 592, 663, 742, 821, 931, 1020, 1147, 1271, 1416, 1558, 1752, 1916, 2137, 2357, 2613, 2867

Offset: 0

Gus Wiseman, Jan 21 2022

nonn

			The a(1) = 1 through a(12) = 11 partitions (A..C = 10..12):
  1   2   3    4   5    6     7    8     9     A      B     C
          21       32   51    43   62    54    73     65    84
                   41   321   52   71    63    82     74    93
                              61   521   72    91     83    A2
                                         81    541    92    B1
                                         432   721    A1    543
                                         621   4321   632   651
                                                      821   732
                                                            741
                                                            921
                                                            6321

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for no difference 0 is A000009.
The version for no difference > -2 is A001227, non-strict A034296.
The version for no difference -1 is A003114 (A325160).
The version for subsets of prescribed maximum is A005314.
The version for all differences < -2 is A025157, non-strict A116932.
The opposite version is A072670.
The multiplicative version is A350840, non-strict A350837 (A350838).
The non-strict version is A350842.
A000041 counts integer partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length (A026424).
A116931 counts partitions with no difference -1 (A319630).
A323092 counts double-free integer partitions (A320340) strict A120641.
A325534 counts separable partitions (A335433).
A325535 counts inseparable partitions (A335448).
Cf. A000929, A003000, A018819, A040039, A045690, A045691, A154402, A303362, A323094, A342095, A342097.

Table[Length[Select[IntegerPartitions[n],FreeQ[Differences[#],0|-2]&]],{n,0,30}]

A350838 Heinz numbers of partitions with no adjacent parts of quotient 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Jan 18 2022

nonn

Differs from A320340 in having 105: (4,3,2), 315: (4,3,2,2), 455: (6,4,3), etc.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with no adjacent prime indices of quotient 1/2.

			The terms and their prime indices begin:
      1: {}            19: {8}             38: {1,8}
      2: {1}           20: {1,1,3}         39: {2,6}
      3: {2}           22: {1,5}           40: {1,1,1,3}
      4: {1,1}         23: {9}             41: {13}
      5: {3}           25: {3,3}           43: {14}
      7: {4}           26: {1,6}           44: {1,1,5}
      8: {1,1,1}       27: {2,2,2}         45: {2,2,3}
      9: {2,2}         28: {1,1,4}         46: {1,9}
     10: {1,3}         29: {10}            47: {15}
     11: {5}           31: {11}            49: {4,4}
     13: {6}           32: {1,1,1,1,1}     50: {1,3,3}
     14: {1,4}         33: {2,5}           51: {2,7}
     15: {2,3}         34: {1,7}           52: {1,1,6}
     16: {1,1,1,1}     35: {3,4}           53: {16}
     17: {7}           37: {12}            55: {3,5}

The version with quotients >= 2 is counted by A000929, sets A018819.
                           <= 2 is A342191, counted by A342094.
                           < 2 is counted by A342096, sets A045690.
                           > 2 is counted by A342098, sets A040039.
The sets version (subsets of prescribed maximum) is counted by A045691.
These partitions are counted by A350837.
The strict case is counted by A350840.
For differences instead of quotients we have A350842, strict A350844.
The complement is A350845, counted by A350846.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A003114 = strict partitions with no successions, ranked by A325160.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A000302, A001105, A003000, A018819, A094537, A120641, A154402, A319613, A323093, A337135, A342097, A342095.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],And@@Table[FreeQ[Divide@@@Partition[primeptn[#],2,1],2],{i,2,PrimeOmega[#]}]&]

A351012 Number of even-length integer partitions y of n such that y_i = y_{i+1} for all even i.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 5, 6, 9, 10, 13, 16, 21, 24, 29, 35, 43, 50, 60, 70, 83, 97, 113, 132, 156, 178, 206, 239, 275, 316, 365, 416, 477, 545, 620, 706, 806, 912, 1034, 1173, 1326, 1496, 1691, 1902, 2141, 2410, 2704, 3034, 3406, 3808, 4261, 4765, 5317, 5932, 6617

Offset: 0

Gus Wiseman, Feb 03 2022

nonn

			The a(2) = 1 through a(8) = 9 partitions:
  (11)  (21)  (22)    (32)    (33)      (43)      (44)
              (31)    (41)    (42)      (52)      (53)
              (1111)  (2111)  (51)      (61)      (62)
                              (3111)    (2221)    (71)
                              (111111)  (4111)    (2222)
                                        (211111)  (3221)
                                                  (5111)
                                                  (311111)
                                                  (11111111)

The ordered version (compositions) is A027383(n-2).
For odd instead of even indices we have A035363, any length A351004.
The version for unequal parts appears to be A122134, any length A122135.
This is the even-length case of A351003.
Requiring inequalities at odd positions gives A351007, any length A351006.
Cf. A000070, A018819, A035457, A088218, A101417, A122129, A350837, A351005, A351008.

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,30}]

A350840 Number of strict integer partitions of n with no adjacent parts of quotient 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 5, 6, 7, 8, 10, 13, 17, 19, 22, 25, 30, 35, 43, 52, 60, 70, 81, 93, 106, 122, 142, 166, 190, 216, 249, 287, 325, 371, 420, 479, 543, 617, 695, 784, 888, 1000, 1126, 1266, 1420, 1594, 1792, 2008, 2247, 2514, 2809, 3135, 3496, 3891, 4332

Offset: 0

Gus Wiseman, Jan 20 2022

nonn

			The a(1) = 1 through a(13) = 13 partitions (A..D = 10..13):
  1   2   3   4    5    6    7    8     9     A     B     C     D
              31   32   51   43   53    54    64    65    75    76
                   41        52   62    72    73    74    93    85
                             61   71    81    82    83    A2    94
                                  431   432   91    92    B1    A3
                                        531   532   A1    543   B2
                                              541   641   651   C1
                                                    731   732   643
                                                          741   652
                                                          831   751
                                                                832
                                                                931
                                                                5431

The version for subsets of prescribed maximum is A045691.
The double-free case is A120641.
The non-strict case is A350837, ranked by A350838.
An additive version (differences) is A350844, non-strict A350842.
The non-strict complement is counted by A350846, ranked by A350845.
Versions for prescribed quotients:
  = 2:  A154402, sets A001511.
  != 2: A350840 (this sequence), sets A045691.
  >= 2: A000929, sets A018819.
  <= 2: A342095, non-strict A342094.
  < 2:  A342097, non-strict A342096, sets A045690.
  > 2:  A342098, sets A040039.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A003114 = strict partitions with no successions, ranked by A325160.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A003000, A018819, A303362, A323093, A323094, A337135, A342191.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[#[[i-1]]/#[[i]]!=2,{i,2,Length[#]}]&]],{n,0,30}]

A045691 Number of binary words of length n with autocorrelation function 2^(n-1)+1.

Original entry on oeis.org

0, 1, 1, 3, 5, 11, 19, 41, 77, 159, 307, 625, 1231, 2481, 4921, 9883, 19689, 39455, 78751, 157661, 315015, 630337, 1260049, 2520723, 5040215, 10081661, 20160841, 40324163, 80643405, 161291731, 322573579, 645157041, 1290294393, 2580608475, 5161177495

Offset: 0

Torsten Sillke (torsten.sillke(AT)lhsystems.com)

nonn

From Gus Wiseman, Jan 22 2022: (Start)
Also the number of subsets of {1..n} containing n but without adjacent elements of quotient 1/2. The Heinz numbers of these sets are a subset of the squarefree terms of A320340. For example, the a(1) = 1 through a(6) = 19 subsets are:
  {1}  {2}  {3}    {4}      {5}        {6}
            {1,3}  {1,4}    {1,5}      {1,6}
            {2,3}  {3,4}    {2,5}      {2,6}
                   {1,3,4}  {3,5}      {4,6}
                   {2,3,4}  {4,5}      {5,6}
                            {1,3,5}    {1,4,6}
                            {1,4,5}    {1,5,6}
                            {2,3,5}    {2,5,6}
                            {3,4,5}    {3,4,6}
                            {1,3,4,5}  {3,5,6}
                            {2,3,4,5}  {4,5,6}
                                       {1,3,4,6}
                                       {1,3,5,6}
                                       {1,4,5,6}
                                       {2,3,4,6}
                                       {2,3,5,6}
                                       {3,4,5,6}
                                       {1,3,4,5,6}
                                       {2,3,4,5,6}
(End)

If a(n) counts subsets of {1..n} with n and without adjacent quotients 1/2:
- The version with quotients <= 1/2 is A018819, partitions A000929.
- The version with quotients < 1/2 is A040039, partitions A342098.
- The version with quotients >= 1/2 is A045690(n+1), partitions A342094.
- The version with quotients > 1/2 is A045690, partitions A342096.
- Partitions of this type are counted by A350837, ranked by A350838.
- Strict partitions of this type are counted by A350840.
- For differences instead of quotients we have A350842, strict A350844.
- Partitions not of this type are counted by A350846, ranked by A350845.
A000740 = relatively prime subsets of {1..n} containing n.
A002843 = compositions with all adjacent quotients >= 1/2.
A050291 = double-free subsets of {1..n}.
A154402 = partitions with all adjacent quotients 2.
A308546 = double-closed subsets of {1..n}, with maximum: shifted right.
A323092 = double-free integer partitions, ranked by A320340, strict A120641.
A326115 = maximal double-free subsets of {1..n}.
Cf. A000009, A001511, A003000, A003114, A116932, A274199, A323093, A342095, A342191, A342331, A342332, A342333, A342337.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&And@@Table[#[[i-1]]/#[[i]]!=1/2,{i,2,Length[#]}]&]],{n,0,15}] (* Gus Wiseman, Jan 22 2022 *)

a(2*n-1) = 2*a(2*n-2) - a(n) for n >= 2; a(2*n) = 2*a(2*n-1) + a(n) for n >= 2.

More terms from Sean A. Irvine, Mar 18 2021

A350845 Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 130, 132, 133, 138, 144, 147, 150, 156, 162, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252, 258, 260, 264, 266, 270

Offset: 1

Gus Wiseman, Jan 20 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with at least two adjacent prime indices of quotient 1/2.

			The terms and corresponding partitions begin:
   6: (2,1)
  12: (2,1,1)
  18: (2,2,1)
  21: (4,2)
  24: (2,1,1,1)
  30: (3,2,1)
  36: (2,2,1,1)
  42: (4,2,1)
  48: (2,1,1,1,1)
  54: (2,2,2,1)
  60: (3,2,1,1)
  63: (4,2,2)
  65: (6,3)
  66: (5,2,1)
  72: (2,2,1,1,1)
  78: (6,2,1)
  84: (4,2,1,1)
  90: (3,2,2,1)
  96: (2,1,1,1,1,1)

The complement is A350838, counted by A350837.
The strict complement is counted by A350840.
These partitions are counted by A350846.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A325160 ranks strict partitions with no successions, counted by A003114.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A000929, A001105, A018819, A045690, A045691, A094537, A154402, A319613, A323093, A337135, A342094, A342095, A342098, A342191.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],MemberQ[Divide@@@Partition[primeptn[#],2,1],2]&]

A350846 Number of integer partitions of n with at least two adjacent parts of quotient 2.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 8, 12, 18, 25, 36, 48, 65, 89, 119, 157, 207, 269, 350, 448, 574, 729, 927, 1166, 1465, 1830, 2282, 2827, 3501, 4309, 5300, 6483, 7923, 9641, 11718, 14187, 17155, 20674, 24885, 29860, 35787, 42772, 51054, 60791, 72289, 85772, 101641

Offset: 0

Gus Wiseman, Jan 20 2022

nonn

			The a(3) = 1 through a(9) = 12 partitions:
  (21)  (211)  (221)   (42)     (421)     (422)      (63)
               (2111)  (321)    (2221)    (521)      (621)
                       (2211)   (3211)    (3221)     (3321)
                       (21111)  (22111)   (4211)     (4221)
                                (211111)  (22211)    (5211)
                                          (32111)    (22221)
                                          (221111)   (32211)
                                          (2111111)  (42111)
                                                     (222111)
                                                     (321111)
                                                     (2211111)
                                                     (21111111)

The complement is counted by A350837, strict A350840.
The complimentary additive version is A350842, strict A350844.
These partitions are ranked by A350845, complement A350838.
A000041 = integer partitions.
A323092 = double-free integer partitions, ranked by A320340.
Cf. A000929, A003000, A003114, A018819, A045690, A045691, A116931, A120641, A154402, A323093, A342094, A342095, A342096, A342098.

Table[Length[Select[IntegerPartitions[n], MemberQ[Divide@@@Partition[#,2,1],2]&]],{n,0,30}]

cp's OEIS Frontend

A350842 Number of integer partitions of n with no difference -2.

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A351003 Number of integer partitions y of n such that y_i = y_{i+1} for all even i.

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A350839 Number of integer partitions of n with a difference < -1 and a conjugate difference < -1.

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A350844 Number of strict integer partitions of n with no difference -2.

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A350838 Heinz numbers of partitions with no adjacent parts of quotient 2.

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A351012 Number of even-length integer partitions y of n such that y_i = y_{i+1} for all even i.

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A350840 Number of strict integer partitions of n with no adjacent parts of quotient 2.

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A045691 Number of binary words of length n with autocorrelation function 2^(n-1)+1.

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A350845 Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.

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