A332281 -id:A332281 - cp's OEIS frontend

A332638 Number of integer partitions of n whose negated run-lengths are unimodal.

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 52, 70, 91, 118, 151, 195, 246, 310, 388, 484, 600, 743, 909, 1113, 1359, 1650, 1996, 2409, 2895, 3471, 4156, 4947, 5885, 6985, 8260, 9751, 11503, 13511, 15857, 18559, 21705, 25304, 29499, 34259, 39785, 46101, 53360, 61594

Offset: 0

Gus Wiseman, Feb 25 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(8) = 21 partitions:
  (8)     (44)     (2222)
  (53)    (332)    (22211)
  (62)    (422)    (32111)
  (71)    (431)    (221111)
  (521)   (3311)   (311111)
  (611)   (4211)   (2111111)
  (5111)  (41111)  (11111111)
Missing from this list is only (3221).

MathWorld, Unimodal Sequence

The non-negated version is A332280.
The complement is counted by A332639.
The Heinz numbers of partitions not in this class are A332642.
The case of 0-appended differences (instead of run-lengths) is A332728.
Unimodal compositions are A001523.
Partitions whose run lengths are not unimodal are A332281.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Compositions whose negation is unimodal are A332578.
Compositions whose run-lengths are unimodal are A332726.
Cf. A007052, A100883, A115981, A181819, A332283, A332577, A332640, A332669, A332670, A332727.

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

A332577 Number of integer partitions of n covering an initial interval of positive integers with unimodal run-lengths.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 25, 30, 36, 40, 45, 54, 59, 68, 79, 86, 96, 112, 121, 135, 155, 168, 188, 214, 230, 253, 284, 308, 337, 380, 407, 445, 497, 533, 580, 645, 689, 748, 828, 885, 956, 1053, 1124, 1212, 1330, 1415, 1519, 1665, 1771

Offset: 0

Gus Wiseman, Feb 24 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(1) = 1 through a(9) = 8 partitions:
  1  11  21   211   221    321     2221     3221      3321
         111  1111  2111   2211    3211     22211     22221
                    11111  21111   22111    32111     32211
                           111111  211111   221111    222111
                                   1111111  2111111   321111
                                            11111111  2211111
                                                      21111111
                                                      111111111

MathWorld, Unimodal Sequence

Not requiring unimodality gives A000009.
A version for compositions is A227038.
Not requiring the partition to cover an initial interval gives A332280.
The complement is counted by A332579.
Unimodal compositions are A001523.
Cf. A007052, A011782, A025065, A100883, A107429, A115981, A332281, A332283, A332638, A332639, A332728.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n],normQ[#]&&unimodQ[Length/@Split[#]]&]],{n,0,30}]

A332745 Number of integer partitions of n whose run-lengths are either weakly increasing or weakly decreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 39, 51, 68, 87, 113, 143, 183, 228, 289, 354, 443, 544, 672, 812, 1001, 1202, 1466, 1758, 2123, 2525, 3046, 3606, 4308, 5089, 6054, 7102, 8430, 9855, 11621, 13571, 15915, 18500, 21673, 25103, 29245, 33835, 39296, 45277, 52470

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

Also partitions whose run-lengths and negated run-lengths are both unimodal.

			The a(8) = 21 partitions are:
  (8)     (44)     (2222)
  (53)    (332)    (22211)
  (62)    (422)    (32111)
  (71)    (431)    (221111)
  (521)   (3311)   (311111)
  (611)   (4211)   (2111111)
  (5111)  (41111)  (11111111)
Missing from this list is only (3221).

The complement is counted by A332641.
The Heinz numbers of partitions not in this class are A332831.
The case of run-lengths of compositions is A332835.
Only weakly decreasing is A100882.
Only weakly increasing is A100883.
Unimodal compositions are A001523.
Non-unimodal compositions are A115981.
Partitions with unimodal run-lengths are A332280.
Partitions whose negated run-lengths are unimodal are A332638.
Compositions with unimodal run-lengths are A332726.
Compositions that are neither weakly increasing nor decreasing are A332834.
Cf. A025065, A059204, A181819, A328509, A332281, A332283, A332577, A332578, A332640, A332727, A332742, A332833.

Table[Length[Select[IntegerPartitions[n],Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,30}]

A332726 Number of compositions of n whose run-lengths are unimodal.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 31, 61, 120, 228, 438, 836, 1580, 2976, 5596, 10440, 19444, 36099, 66784, 123215, 226846, 416502, 763255, 1395952, 2548444, 4644578, 8452200, 15358445, 27871024, 50514295, 91446810, 165365589, 298730375, 539127705, 972099072, 1751284617, 3152475368

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

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

A composition of n is a finite sequence of positive integers summing to n.

			The only composition of 6 whose run-lengths are not unimodal is (1,1,2,1,1).

Andrew Howroyd, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.

Looking at the composition itself (not run-lengths) gives A001523.
The case of partitions is A332280, with complement counted by A332281.
The complement is counted by A332727.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Non-unimodal compositions are A115981.
Compositions with normal run-lengths are A329766.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283, with complement A332284, with Heinz numbers A332287.
Compositions whose negated run-lengths are unimodal are A332578.
Compositions whose negated run-lengths are not unimodal are A332669.
Compositions whose run-lengths are weakly increasing are A332836.
Cf. A072706, A100883, A181819, A227038, A328509, A329744, A329746, A332642, A332670, A332741, A332833, A332835.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],unimodQ[Length/@Split[#]]&]],{n,0,10}]

step(M, m)={my(n=matsize(M)[1]); for(p=m+1, n, my(v=vector((p-1)\m, i, M[p-i*m,i]), s=vecsum(v)); M[p,]+=vector(#M,i,s-if(i<=#v, v[i]))); M}
desc(M, m)={my(n=matsize(M)[1]); while(m>1, m--; M=step(M,m)); vector(n, i, vecsum(M[i,]))/(#M-1)}
seq(n)={my(M=matrix(n+1, n+1, i, j, i==1), S=M[,1]~); for(m=1, n, my(D=M); M=step(M, m); D=(M-D)[m+1..n+1,1..n-m+2]; S+=concat(vector(m), desc(D,m))); S} \\ Andrew Howroyd, Dec 31 2020

a(n) + A332727(n) = 2^(n - 1).

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

A332287 Heinz numbers of integer partitions whose first differences (assuming the last part is zero) are not unimodal.

Original entry on oeis.org

36, 50, 70, 72, 98, 100, 108, 140, 144, 154, 180, 182, 196, 200, 216, 225, 242, 250, 252, 280, 286, 288, 294, 300, 308, 324, 338, 350, 360, 363, 364, 374, 392, 396, 400, 418, 429, 432, 441, 442, 450, 462, 468, 484, 490, 494, 500, 504, 507, 540, 550, 560, 561

Offset: 1

Gus Wiseman, Feb 21 2020

nonn

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), which gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
   36: {1,1,2,2}
   50: {1,3,3}
   70: {1,3,4}
   72: {1,1,1,2,2}
   98: {1,4,4}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  140: {1,1,3,4}
  144: {1,1,1,1,2,2}
  154: {1,4,5}
  180: {1,1,2,2,3}
  182: {1,4,6}
  196: {1,1,4,4}
  200: {1,1,1,3,3}
  216: {1,1,1,2,2,2}
  225: {2,2,3,3}
  242: {1,5,5}
  250: {1,3,3,3}
  252: {1,1,2,2,4}
  280: {1,1,1,3,4}
For example, the prime indices of 70 with 0 appended are (4,3,1,0), with differences (-1,-2,-1), which is not unimodal, so 70 belongs to the sequence.

MathWorld, Unimodal Sequence

The enumeration of these partitions by sum is A332284.
Not assuming the last part is zero gives A332725.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Cf. A001523, A007052, A332280, A332282, A332283, A332285, A332286, A332288, A332294, A332579, A332639, A332642.

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[1000],!unimodQ[Differences[Append[Reverse[primeMS[#]],0]]]&]

A332870 Number of compositions of n that are neither unimodal nor is their negation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 9, 32, 92, 243, 587, 1361, 3027, 6564, 13928, 29127, 60180, 123300, 250945, 508326, 1025977, 2065437, 4150056, 8327344, 16692844, 33438984, 66951671, 134004892, 268148573, 536486146, 1073227893, 2146800237, 4294061970, 8588740071, 17178298617

Offset: 0

Gus Wiseman, Mar 02 2020

nonn

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

A composition of n is a finite sequence of positive integers summing to n.

			The a(6) = 2 and a(7) = 9 compositions:
  (1212)  (1213)
  (2121)  (1312)
          (2131)
          (3121)
          (11212)
          (12112)
          (12121)
          (21121)
          (21211)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The case of run-lengths of partitions is A332640.
The version for unsorted prime signature is A332643.
Unimodal compositions are A001523.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Compositions whose negation is unimodal are A332578.
Compositions whose negation is not unimodal are A332669.
Partitions with weakly increasing or decreasing run-lengths are A332745.
Compositions that are neither weakly increasing nor decreasing are A332834.
Compositions with weakly increasing or decreasing run-lengths are A332835.
Cf. A000005, A000041, A007052, A072704, A227038, A329398, A332281, A332284, A332639, A332641, A332746, A332831, A332833.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!unimodQ[#]&&!unimodQ[-#]&]],{n,0,10}]

a(n) = 2^(n-1) - A001523(n) - A332578(n) + 2*A000041(n) - A000005(n) for n > 0. - Andrew Howroyd, Dec 30 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A332286 Number of strict integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 3, 5, 5, 7, 9, 12, 15, 22, 23, 31, 40, 47, 58, 72, 81, 100, 122, 144, 171, 206, 236, 280, 333, 381, 445, 522, 593, 694, 802, 914, 1054, 1214, 1376, 1577, 1803, 2040, 2324, 2646, 2973, 3373, 3817, 4287, 4838, 5453, 6096, 6857

Offset: 0

Gus Wiseman, Feb 21 2020

nonn

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

Also the number integer partitions of n that cover an initial interval of positive integers and whose negated run-lengths are not unimodal.

			The a(8) = 1 through a(18) = 7 partitions:
  (431)  .  (541)  (641)  (651)   (652)   (752)   (762)   (862)
                          (5421)  (751)   (761)   (861)   (871)
                                  (5431)  (851)   (6531)  (961)
                                          (6431)  (7431)  (6532)
                                          (6521)  (7521)  (6541)
                                                          (7621)
                                                          (8431)
For example, (4,3,1,0) has first differences (-1,-2,-1), which is not unimodal, so (4,3,1) is counted under a(8).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.

Strict partitions are A000009.
Partitions covering an initial interval are (also) A000009.
The non-strict version is A332284.
The complement is counted by A332285.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Normal partitions whose run-lengths are not unimodal are A332579.
Cf. A007052, A011782, A025065, A072706, A227038, A332282, A332283, A332286, A332287, A332288, A332577, A332638, A332642, A332743.

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

A332640 Number of integer partitions of n such that neither the run-lengths nor the negated run-lengths are unimodal.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 4, 6, 12, 17, 29, 44, 66, 92, 138, 187, 266, 359, 492, 649, 877, 1140, 1503, 1938, 2517, 3202, 4111, 5175, 6563, 8209, 10297, 12763, 15898, 19568, 24152, 29575, 36249, 44090, 53737, 65022, 78752, 94873, 114294

Offset: 0

Gus Wiseman, Feb 25 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(14) = 1 through a(18) = 12 partitions:
  (433211)  (533211)   (443221)    (544211)     (544311)
            (4332111)  (633211)    (733211)     (553221)
                       (5332111)   (4333211)    (644211)
                       (43321111)  (6332111)    (833211)
                                   (53321111)   (4432221)
                                   (433211111)  (5333211)
                                                (5442111)
                                                (7332111)
                                                (43332111)
                                                (63321111)
                                                (533211111)
                                                (4332111111)
For example, the partition (4,3,3,2,1,1) has run-lengths (1,2,1,2), so is counted under a(14).

MathWorld, Unimodal Sequence

Looking only at the original run-lengths gives A332281.
Looking only at the negated run-lengths gives A332639.
The Heinz numbers of these partitions are A332643.
The complement is counted by A332746.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Partitions with unimodal run-lengths are A332280.
Partitions whose negated run-lengths are unimodal are A332638.
Run-lengths and negated run-lengths are not both unimodal: A332641.
Compositions whose negation is not unimodal are A332669.
Run-lengths and negated run-lengths are both unimodal: A332745.
Cf. A007052, A025065, A100883, A181819, A328509, A332282, A332284, A332577, A332578, A332579, A332642, A332726, A332727.

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

A332831 Numbers whose unsorted prime signature is neither weakly increasing nor weakly decreasing.

Original entry on oeis.org

90, 126, 198, 234, 270, 300, 306, 342, 350, 378, 414, 522, 525, 540, 550, 558, 588, 594, 600, 630, 650, 666, 702, 738, 756, 774, 810, 825, 846, 850, 918, 950, 954, 975, 980, 990, 1026, 1050, 1062, 1078, 1098, 1134, 1150, 1170, 1176, 1188, 1200, 1206, 1242

Offset: 1

Gus Wiseman, Mar 02 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The sequence of terms together with their prime indices begins:
   90: {1,2,2,3}
  126: {1,2,2,4}
  198: {1,2,2,5}
  234: {1,2,2,6}
  270: {1,2,2,2,3}
  300: {1,1,2,3,3}
  306: {1,2,2,7}
  342: {1,2,2,8}
  350: {1,3,3,4}
  378: {1,2,2,2,4}
  414: {1,2,2,9}
  522: {1,2,2,10}
  525: {2,3,3,4}
  540: {1,1,2,2,2,3}
  550: {1,3,3,5}
  558: {1,2,2,11}
  588: {1,1,2,4,4}
  594: {1,2,2,2,5}
  600: {1,1,1,2,3,3}
  630: {1,2,2,3,4}
For example, the prime signature of 540 is (2,3,1), so 540 is in the sequence.

The version for run-lengths of partitions is A332641.
The version for run-lengths of compositions is A332833.
The version for compositions is A332834.
Prime signature is A124010.
Unimodal compositions are A001523.
Partitions with weakly increasing run-lengths are A100883.
Partitions with weakly increasing or decreasing run-lengths are A332745.
Compositions with weakly increasing or decreasing run-lengths are A332835.
Compositions with weakly increasing run-lengths are A332836.
Cf. A100882, A115981, A328509, A329398, A332281, A332640, A332643, A332746, A332836, A332870.

Select[Range[1000],!Or[LessEqual@@Last/@FactorInteger[#],GreaterEqual@@Last/@FactorInteger[#]]&]

Intersection of A071365 and A112769.

A332641 Number of integer partitions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 9, 14, 22, 33, 48, 69, 96, 136, 184, 248, 330, 443, 574, 756, 970, 1252, 1595, 2040, 2558, 3236, 4041, 5054, 6256, 7781, 9547, 11782, 14394, 17614, 21423, 26083, 31501, 38158, 45930, 55299, 66262, 79477, 94803, 113214

Offset: 0

Gus Wiseman, Feb 26 2020

nonn

Also partitions whose run-lengths and negated run-lengths are not both unimodal. A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

			The a(8) = 1 through a(13) = 14 partitions:
  (3221)  (4221)  (5221)   (4331)    (4332)     (5332)
                  (32221)  (6221)    (5331)     (6331)
                  (33211)  (42221)   (7221)     (8221)
                           (322211)  (43221)    (43321)
                           (332111)  (44211)    (44311)
                                     (52221)    (53221)
                                     (322221)   (62221)
                                     (422211)   (332221)
                                     (3321111)  (333211)
                                                (422221)
                                                (442111)
                                                (522211)
                                                (3222211)
                                                (33211111)

MathWorld, Unimodal Sequence

The complement is counted by A332745.
The Heinz numbers of these partitions are A332831.
The case of run-lengths of compositions is A332833.
Partitions whose run-lengths are weakly increasing are A100883.
Partitions whose run-lengths are weakly decreasing are A100882.
Partitions whose run-lengths are not unimodal are A332281.
Partitions whose negated run-lengths are not unimodal are A332639.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Partitions with unimodal run-lengths are A332280.
Partitions whose negated run-lengths are unimodal are A332638.
Compositions whose negation is not unimodal are A332669.
The case of run-lengths of compositions is A332833.
Compositions that are neither increasing nor decreasing are A332834.
Cf. A025065, A181819, A328509, A332282, A332284, A332577, A332578, A332579, A332640, A332642, A332726, A332727, A332742, A332835.

Table[Length[Select[IntegerPartitions[n],!Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,30}]

cp's OEIS Frontend

A332638 Number of integer partitions of n whose negated run-lengths are unimodal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A332577 Number of integer partitions of n covering an initial interval of positive integers with unimodal run-lengths.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A332745 Number of integer partitions of n whose run-lengths are either weakly increasing or weakly decreasing.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A332726 Number of compositions of n whose run-lengths are unimodal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A332287 Heinz numbers of integer partitions whose first differences (assuming the last part is zero) are not unimodal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A332870 Number of compositions of n that are neither unimodal nor is their negation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A332286 Number of strict integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A332640 Number of integer partitions of n such that neither the run-lengths nor the negated run-lengths are unimodal.

Views

Author

Keywords

Comments