A332638 -id:A332638 - cp's OEIS frontend

A332728 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are unimodal.

1, 1, 2, 3, 4, 5, 7, 8, 10, 13, 14, 17, 22, 24, 28, 34, 37, 43, 53, 56, 64, 76, 83, 93, 111, 117, 131, 153, 163, 182, 210, 225, 250, 284, 304, 332, 377, 401, 441, 497, 529, 576, 647, 687, 745, 830, 883, 955, 1062, 1127, 1216, 1339, 1422, 1532, 1684, 1779, 1914

Offset: 0

Gus Wiseman, Feb 26 2020

nonn

First differs from A000041 at a(6) = 10, A000041(6) = 11.

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

			The a(1) = 1 through a(8) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (421)      (332)
                                     (111111)  (2221)     (431)
                                               (1111111)  (521)
                                                          (2222)
                                                          (11111111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..600
Eric Weisstein's World of Mathematics, Unimodal Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

The non-negated version is A332283.
The non-negated complement is counted by A332284.
The strict case is A332577.
The case of run-lengths (instead of differences) is A332638.
The complement is counted by A332744.
The Heinz numbers of partitions not in this class are A332287.
Unimodal compositions are A001523.
Compositions whose negation is unimodal are A332578.
Compositions whose run-lengths are unimodal are A332726.
Cf. A007052, A332280, A332285, A332286, A332639, A332642, A332669, A332670, A332741.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n],unimodQ[-Differences[Append[#,0]]]&]],{n,0,30}]

A332741 Number of unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 3, 8, 1, 6, 1, 4, 3, 2, 1, 8, 4, 2, 9, 4, 1, 6, 1, 16, 3, 2, 4, 12, 1, 2, 3, 8, 1, 6, 1, 4, 9, 2, 1, 16, 5, 8, 3, 4, 1, 18, 4, 8, 3, 2, 1, 12, 1, 2, 9, 32, 4, 6, 1, 4, 3, 8, 1, 24, 1, 2, 12, 4, 5, 6, 1, 16, 27, 2, 1

Offset: 1

Gus Wiseman, Mar 09 2020

nonn

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

			The a(12) = 4 permutations:
  {1,1,2,3}
  {2,1,1,3}
  {3,1,1,2}
  {3,2,1,1}

Eric Weisstein's World of Mathematics, Unimodal Sequence

Dominated by A318762.
The non-negated version is A332294.
The complement is counted by A332742.
A less interesting version is A333145.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Numbers with non-unimodal negated prime signature are A332642.
Partitions whose 0-appended first differences are unimodal are A332283.
Compositions whose negation is unimodal are A332578.
Partitions with unimodal negated run-lengths are A332638.
Cf. A056239, A112798, A115981, A124010, A181819, A181821, A304660, A332280, A332288, A332639, A332669, A332672.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[nrmptn[n]],unimodQ[-#]&]],{n,30}]

a(n) + A332742(n) = A318762(n).

A332744 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 4, 7, 12, 17, 28, 39, 55, 77, 107, 142, 194, 254, 332, 434, 563, 716, 919, 1162, 1464, 1841, 2305, 2857, 3555, 4383, 5394, 6617, 8099, 9859, 12006, 14551, 17600, 21236, 25574, 30688, 36809, 44007, 52527, 62574, 74430, 88304, 104675, 123799

Offset: 0

Gus Wiseman, Feb 27 2020

nonn

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

			The a(4) = 1 through a(9) = 17 partitions:
  (211)  (311)   (411)    (322)     (422)      (522)
         (2111)  (2211)   (511)     (611)      (711)
                 (3111)   (3211)    (3221)     (3222)
                 (21111)  (4111)    (3311)     (4221)
                          (22111)   (4211)     (4311)
                          (31111)   (5111)     (5211)
                          (211111)  (22211)    (6111)
                                    (32111)    (32211)
                                    (41111)    (33111)
                                    (221111)   (42111)
                                    (311111)   (51111)
                                    (2111111)  (222111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
For example, the partition y = (4,2,1,1,1) has negated 0-appended first differences (2,1,0,0,1), which is not unimodal, so y is counted under a(9).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..600
Eric Weisstein's World of Mathematics, Unimodal Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

The complement is counted by A332728.
The non-negated version is A332284.
The strict case is A332579.
The case of run-lengths (instead of differences) is A332639.
The Heinz numbers of these partitions are A332832.
Unimodal compositions are A001523.
Non-unimodal compositions are A115981.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Partitions whose 0-appended first differences are unimodal are A332283.
Compositions whose negation is unimodal are A332578.
Numbers whose negated prime signature is not unimodal are A332642.
Compositions whose negation is not unimodal are A332669.
Cf. A059204, A227038, A332280, A332285, A332286, A332287, A332638, A332670, A332725, A332726.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n],!unimodQ[-Differences[Append[#,0]]]&]],{n,0,30}]

A332832 Heinz numbers of integer partitions whose negated first differences (assuming the last part is zero) are not unimodal.

Original entry on oeis.org

12, 20, 24, 28, 36, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 165, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 195, 196, 198

Offset: 1

Gus Wiseman, Mar 02 2020

nonn

First differs from A065201 in having 165.
First differs from A316597 in having 36.
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   68: {1,1,7}
   72: {1,1,1,2,2}
   76: {1,1,8}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   88: {1,1,1,5}
   90: {1,2,2,3}
For example, 60 is the Heinz number of (3,2,1,1), with negated 0-appended first-differences (1,1,0,1), which are not unimodal, so 60 is in the sequence.

MathWorld, Unimodal Sequence
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

The non-negated version is A332287.
The version for of run-lengths (instead of differences) is A332642.
The enumeration of these partitions by sum is A332744.
Unimodal compositions are A001523.
Non-unimodal compositions are A115981.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Partitions whose 0-appended first differences are unimodal are A332283.
Compositions whose negation is unimodal are A332578.
Compositions whose negation is not unimodal are A332669.
Cf. A059204, A227038, A332284, A332285, A332286, A332578, A332638, A332639, A332670, A332725, A332728.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Select[Range[100],!unimodQ[Differences[Prepend[primeMS[#],0]]]&]

A333146 Number of non-unimodal negated permutations of the multiset of prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 8, 0, 0, 1, 0, 0, 2, 0, 1, 0, 2, 0, 7, 0, 0, 0, 1, 0, 2, 0, 3, 0, 0, 0, 8, 0, 0, 0

Offset: 1

Gus Wiseman, Mar 09 2020

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.

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

			The a(n) permutations for n = 12, 24, 36, 60, 72, 90, 96:
  (121)  (1121)  (1212)  (1132)  (11212)  (1232)  (111121)
         (1211)  (1221)  (1213)  (11221)  (1322)  (111211)
                 (2121)  (1231)  (12112)  (2132)  (112111)
                         (1312)  (12121)  (2231)  (121111)
                         (1321)  (12211)  (2312)
                         (2131)  (21121)  (2321)
                         (2311)  (21211)
                         (3121)

Eric Weisstein's World of Mathematics, Unimodal Sequence

Dominated by A008480.
The non-negated version is A332671.
A more interesting version is A332742.
The complement is counted by A333145.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Compositions whose negation is unimodal are A332578.
Partitions with unimodal negated run-lengths are A332638.
Numbers with non-unimodal negated unsorted prime signature are A332642.
Cf. A056239, A112798, A115981, A124010, A328509, A332283, A332288, A332294, A332639, A332669, A332670, A332741.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[primeMS[n]],!unimodQ[-#]&]],{n,30}]

a(n) + A333145(n) = A008480(n).

A335374 Numbers k such that the k-th composition in standard order (A066099) is not co-unimodal.

Original entry on oeis.org

13, 25, 27, 29, 41, 45, 49, 50, 51, 53, 54, 55, 57, 59, 61, 77, 81, 82, 83, 89, 91, 93, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 123, 125, 141, 145, 153, 155, 157, 161, 162, 163, 165, 166, 167, 169, 173, 177

Offset: 1

Gus Wiseman, Jun 03 2020

nonn

A sequence of integers is co-unimodal if it is the concatenation of a weakly decreasing and a weakly increasing sequence, implying that its negation is unimodal.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding compositions begins:
  13: (1,2,1)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  41: (2,3,1)
  45: (2,1,2,1)
  49: (1,4,1)
  50: (1,3,2)
  51: (1,3,1,1)
  53: (1,2,2,1)
  54: (1,2,1,2)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)
  61: (1,1,1,2,1)
  77: (3,1,2,1)
  81: (2,4,1)
  82: (2,3,2)
  83: (2,3,1,1)
  89: (2,1,3,1)

This is the dual version of A335373.
The case that is not unimodal either is A335375.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Unimodal permutations are A011782.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers with non-unimodal unsorted prime signature are A332282.
Co-unimodal compositions are A332578.
Numbers with non-co-unimodal unsorted prime signature are A332642.
Non-co-unimodal compositions are A332669.
Cf. A000120, A029931, A048793, A066099, A070939, A334299.
Cf. A112798, A227038, A329398, A332281, A332286, A332287, A332638, A332639, A332643, A332670, A332873, A333146.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!unimodQ[-stc[#]]&]

A333145 Number of unimodal negated permutations of the multiset of prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 4, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 4, 1, 2, 2, 4, 1, 3, 1, 2, 3, 2, 2, 4, 1, 2, 1, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 09 2020

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.
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
Also permutations of the multiset of prime indices of n avoiding the patterns (1,2,1), (1,3,2), and (2,3,1).
Also the number divisors of n not divisible by the least prime factor of n. The other divisors are counted by A069157. - Gus Wiseman, Apr 12 2022

			The a(n) permutations for n = 2, 6, 18, 30, 90, 162, 210, 450:
  (1)  (12)  (122)  (123)  (1223)  (12222)  (1234)  (12233)
       (21)  (212)  (213)  (2123)  (21222)  (2134)  (21233)
             (221)  (312)  (2213)  (22122)  (3124)  (22133)
                    (321)  (3122)  (22212)  (3214)  (31223)
                           (3212)  (22221)  (4123)  (32123)
                           (3221)           (4213)  (32213)
                                            (4312)  (33122)
                                            (4321)  (33212)
                                                    (33221)

Eric Weisstein's World of Mathematics, Unimodal Sequence
Wikipedia, Permutation pattern

Dominated by A008480.
The complementary divisors are counted by A069157.
The non-negated version is A332288.
A more interesting version is A332741.
The complement is counted by A333146.
A001523 counts unimodal compositions.
A007052 counts unimodal normal sequences.
A028233 gives the highest power of the least prime factor, quotient A028234.
A332578 counts compositions whose negation is unimodal.
A332638 counts partitions with unimodal negated run-lengths.
A332642 lists numbers with non-unimodal negated unsorted prime signature.
Cf. A056239, A112798, A115981, A124010, A328509, A332283, A332294, A332639, A332669, A332670, A332671.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[primeMS[n]],unimodQ[-#]&]],{n,30}]

a(n) + A333146(n) = A008480(n).
a(n) = A000005(A028234(n)). - Gus Wiseman, Apr 14 2022
a(n) = A000005(n) - A069157(n). - Gus Wiseman, Apr 14 2022

A335375 Numbers k such that the k-th composition in standard order (A066099) is neither unimodal nor co-unimodal.

Original entry on oeis.org

45, 54, 77, 89, 91, 93, 102, 108, 109, 110, 118, 141, 153, 155, 157, 166, 173, 177, 178, 179, 181, 182, 183, 185, 187, 189, 198, 204, 205, 206, 214, 216, 217, 218, 219, 220, 221, 222, 230, 236, 237, 238, 246, 269, 281, 283, 285, 297, 301, 305, 306, 307, 309

Offset: 1

Gus Wiseman, Jun 04 2020

nonn

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence. It is co-unimodal if its negation is unimodal.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding compositions begins:
   45: (2,1,2,1)
   54: (1,2,1,2)
   77: (3,1,2,1)
   89: (2,1,3,1)
   91: (2,1,2,1,1)
   93: (2,1,1,2,1)
  102: (1,3,1,2)
  108: (1,2,1,3)
  109: (1,2,1,2,1)
  110: (1,2,1,1,2)
  118: (1,1,2,1,2)
  141: (4,1,2,1)
  153: (3,1,3,1)
  155: (3,1,2,1,1)
  157: (3,1,1,2,1)
  166: (2,3,1,2)
  173: (2,2,1,2,1)
  177: (2,1,4,1)
  178: (2,1,3,2)
  179: (2,1,3,1,1)

Non-unimodal compositions are ranked by A335373.
Non-co-unimodal compositions are ranked by A335374.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Unimodal permutations are A011782.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers with non-unimodal unsorted prime signature are A332282.
Co-unimodal compositions are A332578.
Numbers with non-co-unimodal unsorted prime signature are A332642.
Non-co-unimodal compositions are A332669.
Cf. A000120, A029931, A048793, A066099, A070939, A334299.
Cf. A112798, A227038, A329398, A332281, A332286, A332287, A332638, A332639, A332643, A332670, A332873, A333146.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!unimodQ[stc[#]]&&!unimodQ[-stc[#]]&]

cp's OEIS Frontend

A332728 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are unimodal.

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A332741 Number of unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.

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A332744 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.

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A332832 Heinz numbers of integer partitions whose negated first differences (assuming the last part is zero) are not unimodal.

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A333146 Number of non-unimodal negated permutations of the multiset of prime indices of n.

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A335374 Numbers k such that the k-th composition in standard order (A066099) is not co-unimodal.

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A333145 Number of unimodal negated permutations of the multiset of prime indices of n.

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A335375 Numbers k such that the k-th composition in standard order (A066099) is neither unimodal nor co-unimodal.

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