cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A332578 Number of compositions of n whose negation is unimodal.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 21, 36, 57, 91, 140, 217, 323, 485, 711, 1039, 1494, 2144, 3032, 4279, 5970, 8299, 11438, 15708, 21403, 29065, 39218, 52725, 70497, 93941, 124562, 164639, 216664, 284240, 371456, 484004, 628419, 813669, 1050144, 1351757, 1734873, 2221018, 2835613
Offset: 0

Views

Author

Gus Wiseman, Feb 28 2020

Keywords

Comments

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.

Examples

			The a(1) = 1 through a(5) = 13 compositions:
  (1)  (2)   (3)    (4)     (5)
       (11)  (12)   (13)    (14)
             (21)   (22)    (23)
             (111)  (31)    (32)
                    (112)   (41)
                    (211)   (113)
                    (1111)  (122)
                            (212)
                            (221)
                            (311)
                            (1112)
                            (2111)
                            (11111)

Links

Crossrefs

Dominated by A001523 (unimodal compositions).
The strict case is A072706.
The case that is unimodal also is A329398.
The complement is counted by A332669.
Row sums of A332670.
Unimodal normal sequences appear to be A007052.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions whose run-lengths are unimodal are A332280.
Partitions whose negated run-lengths are unimodal are A332638.
Numbers whose unsorted prime signature is not unimodal are A332642.
Partitions whose negated 0-appended differences are unimodal are A332728.
Cf. A011782, A072704, A107429, A227038, A332282, A332283, A332639, A332741, A332742, A332744, A332832, A332870.

Programs

  • Mathematica
    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[-#]&]],{n,0,10}]
nmax = 50; CoefficientList[Series[1 + Sum[x^j*(1 - x^j)/Product[1 - x^k, {k, j, nmax - j}]^2, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 01 2020 *)
  • PARI
    seq(n)={Vec(1 + sum(j=1, n, x^j/((1-x^j)*prod(k=j+1, n-j, 1 - x^k + O(x*x^(n-j)))^2)))} \\ Andrew Howroyd, Mar 01 2020

Formula

a(n) + A332669(n) = 2^(n - 1).
G.f.: 1 + Sum_{j>0} x^j/((1 - x^j)*(Product_{k>j} 1 - x^k)^2). - Andrew Howroyd, Mar 01 2020
a(n) ~ Pi * exp(2*Pi*sqrt(n/3)) / (4 * 3^(5/4) * n^(7/4)). - Vaclav Kotesovec, Mar 01 2020

Extensions

Terms a(26) and beyond from Andrew Howroyd, Mar 01 2020

A332669 Number of compositions of n whose negation is not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 11, 28, 71, 165, 372, 807, 1725, 3611, 7481, 15345, 31274, 63392, 128040, 257865, 518318, 1040277, 2085714, 4178596, 8367205, 16748151, 33515214, 67056139, 134147231, 268341515, 536746350, 1073577185, 2147266984, 4294683056, 8589563136, 17179385180
Offset: 0

Views

Author

Gus Wiseman, Feb 28 2020

Keywords

Comments

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.

Examples

			The a(4) = 1 through a(6) = 11 compositions:
  (121)  (131)   (132)
         (1121)  (141)
         (1211)  (231)
                 (1131)
                 (1212)
                 (1221)
                 (1311)
                 (2121)
                 (11121)
                 (11211)
                 (12111)

Links

Crossrefs

The strict case is A072707.
The complement is counted by A332578.
The version for run-lengths of partitions is A332639.
The version for unsorted prime signature is A332642.
The version for 0-appended first-differences of partitions is A332744.
The case that is not unimodal either is A332870.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers whose unsorted prime signature is not unimodal are A332282.
A triangle for compositions with unimodal negation is A332670.
Cf. A007052, A072706, A227038, A329398, A332281, A332284, A332638, A332728, A332742, A332832.

Programs

  • Mathematica
    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[-#]&]],{n,0,10}]

Formula

a(n) + A332578(n) = 2^(n - 1) for n > 0.

Extensions

Terms a(21) and beyond from Andrew Howroyd, Mar 01 2020

A332639 Number of integer partitions of n whose negated run-lengths are not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 7, 10, 17, 25, 36, 51, 75, 102, 143, 192, 259, 346, 462, 599, 786, 1014, 1309, 1670, 2133, 2686, 3402, 4258, 5325, 6623, 8226, 10134, 12504, 15328, 18779, 22878, 27870, 33762, 40916, 49349, 59457, 71394, 85679, 102394
Offset: 0

Views

Author

Gus Wiseman, Feb 25 2020

Keywords

Comments

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

Examples

			The a(8) = 1 through a(13) = 10 partitions:
  (3221)  (4221)  (5221)   (4331)    (4332)    (5332)
                  (32221)  (6221)    (5331)    (6331)
                           (42221)   (7221)    (8221)
                           (322211)  (43221)   (43321)
                                     (52221)   (53221)
                                     (322221)  (62221)
                                     (422211)  (332221)
                                               (422221)
                                               (522211)
                                               (3222211)

Links

Crossrefs

The version for normal sequences is A328509.
The non-negated complement is A332280.
The non-negated version is A332281.
The complement is counted by A332638.
The case that is not unimodal either is A332640.
The Heinz numbers of these partitions are A332642.
The generalization to run-lengths of compositions is A332727.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Compositions whose negation is not unimodal are A332669.
Cf. A007052, A025065, A100883, A181819, A332282, A332578, A332579, A332641, A332670, A332671, A332726, A332742, A332744.

Programs

  • Mathematica
    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}]

A332642 Numbers whose negated unsorted prime signature is not unimodal.

Original entry on oeis.org

90, 126, 198, 234, 270, 306, 342, 350, 378, 414, 522, 525, 540, 550, 558, 594, 630, 650, 666, 702, 738, 756, 774, 810, 825, 846, 850, 918, 950, 954, 975, 990, 1026, 1050, 1062, 1078, 1098, 1134, 1150, 1170, 1188, 1206, 1242, 1274, 1275, 1278, 1314, 1350, 1386
Offset: 1

Views

Author

Gus Wiseman, Feb 28 2020

Keywords

Comments

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Examples

			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}
   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}
   594: {1,2,2,2,5}
   630: {1,2,2,3,4}
   650: {1,3,3,6}
   666: {1,2,2,12}
   702: {1,2,2,2,6}
For example, 630 has negated unsorted prime signature (-1,-2,-1,-1), which is not unimodal, so 630 is in the sequence.

Links

Crossrefs

These are the Heinz numbers of the partitions counted by A332639.
The case that is not unimodal either is A332643.
The version for compositions is A332669.
The complement is A332282.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Unsorted prime signature is A124010.
Non-unimodal normal sequences are A328509.
The number of non-unimodal negated permutations of a multiset whose multiplicities are the prime indices of n is A332742(n).
Partitions whose negated 0-appended first differences are not unimodal are A332744, with Heinz numbers A332832.
Cf. A007052, A056239, A112798, A181821, A242031, A329747, A332280, A332281, A332578, A332671, A332831.

Programs

  • Mathematica
    unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Select[Range[2000],!unimodQ[-Last/@FactorInteger[#]]&]

A332670 Triangle read by rows where T(n,k) is the number of length-k compositions of n whose negation is unimodal.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 5, 2, 1, 0, 1, 5, 7, 5, 2, 1, 0, 1, 6, 11, 10, 5, 2, 1, 0, 1, 7, 15, 16, 10, 5, 2, 1, 0, 1, 8, 20, 24, 20, 10, 5, 2, 1, 0, 1, 9, 25, 36, 31, 20, 10, 5, 2, 1, 0, 1, 10, 32, 50, 50, 36, 20, 10, 5, 2, 1
Offset: 0

Views

Author

Gus Wiseman, Feb 29 2020

Keywords

Comments

A composition of n is a finite sequence of positive integers summing to n.
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

Examples

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  2  1
  0  1  3  2  1
  0  1  4  5  2  1
  0  1  5  7  5  2  1
  0  1  6 11 10  5  2  1
  0  1  7 15 16 10  5  2  1
  0  1  8 20 24 20 10  5  2  1
  0  1  9 25 36 31 20 10  5  2  1
  0  1 10 32 50 50 36 20 10  5  2  1
  0  1 11 38 67 73 59 36 20 10  5  2  1
Column n = 7 counts the following compositions:
  (7)  (16)  (115)  (1114)  (11113)  (111112)  (1111111)
       (25)  (124)  (1123)  (11122)  (211111)
       (34)  (133)  (1222)  (21112)
       (43)  (214)  (2113)  (22111)
       (52)  (223)  (2122)  (31111)
       (61)  (313)  (2212)
             (322)  (2221)
             (331)  (3112)
             (412)  (3211)
             (421)  (4111)
             (511)

Links

Crossrefs

The case of partitions is A072233.
Dominated by A072704 (the non-negated version).
The strict case is A072705.
The case of constant compositions is A113704.
Row sums are A332578.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers whose negated unsorted prime signature is not unimodal are A332282.
Partitions whose negated run-lengths are unimodal are A332638.
Compositions whose negation is not unimodal are A332669.
Partitions whose negated 0-appended first differences are unimodal: A332728.
Cf. A011782, A107429, A227038, A332280, A332283, A332639, A332642, A332741, A332742, A332744, A332832, A332870.

Programs

  • Mathematica
    unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],unimodQ[-#]&]],{n,0,10},{k,0,n}]
  • PARI
    T(n)={[Vecrev(p) | p<-Vec(1 + sum(j=1, n, y*x^j/((1-y*x^j) * prod(k=j+1, n-j, 1 - y*x^k + O(x*x^(n-j)))^2)))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

Formula

G.f.: A(x,y) = 1 + Sum_{j>0} y*x^j/((1 - y*x^j)*Product_{k>j} (1 - y*x^k)^2). - Andrew Howroyd, Jan 11 2024

A332643 Neither the unsorted prime signature of a(n) nor the negated unsorted prime signature of a(n) is unimodal.

Original entry on oeis.org

2100, 3300, 3900, 4200, 4410, 5100, 5700, 6468, 6600, 6900, 7644, 7800, 8400, 8700, 9300, 9996, 10200, 10500, 10780, 10890, 11100, 11172, 11400, 12300, 12740, 12900, 12936, 13200, 13230, 13524, 13800, 14100, 15210, 15246, 15288, 15600, 15900, 16500, 16660
Offset: 1

Views

Author

Gus Wiseman, Feb 28 2020

Keywords

Comments

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Examples

			The sequence of terms together with their prime indices begins:
   2100: {1,1,2,3,3,4}
   3300: {1,1,2,3,3,5}
   3900: {1,1,2,3,3,6}
   4200: {1,1,1,2,3,3,4}
   4410: {1,2,2,3,4,4}
   5100: {1,1,2,3,3,7}
   5700: {1,1,2,3,3,8}
   6468: {1,1,2,4,4,5}
   6600: {1,1,1,2,3,3,5}
   6900: {1,1,2,3,3,9}
   7644: {1,1,2,4,4,6}
   7800: {1,1,1,2,3,3,6}
   8400: {1,1,1,1,2,3,3,4}
   8700: {1,1,2,3,3,10}
   9300: {1,1,2,3,3,11}
   9996: {1,1,2,4,4,7}
  10200: {1,1,1,2,3,3,7}
  10500: {1,1,2,3,3,3,4}
  10780: {1,1,3,4,4,5}
  10890: {1,2,2,3,5,5}

Links

Crossrefs

Not requiring non-unimodal negation gives A332282.
These are the Heinz numbers of the partitions counted by A332640.
Not requiring non-unimodality gives A332642.
The case of compositions is A332870.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Unsorted prime signature is A124010.
Non-unimodal normal sequences are A328509.
Partitions whose 0-appended first differences are unimodal are A332283, with Heinz numbers the complement of A332287.
Compositions whose negation is unimodal are A332578.
Compositions whose negation is not unimodal are A332669.
Partitions whose 0-appended first differences are not unimodal are A332744, with Heinz numbers A332832.
Numbers whose signature is neither increasing nor decreasing are A332831.
Cf. A007052, A056239, A072704, A112798, A242031, A242414, A332280, A332281, A332288, A332294, A332639, A332728, A332742.

Programs

  • Mathematica
    unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Select[Range[10000],!unimodQ[Last/@FactorInteger[#]]&&!unimodQ[-Last/@FactorInteger[#]]&]

Formula

Intersection of A332282 and A332642.

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Feb 26 2020

Keywords

Comments

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.

Examples

			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)

Links

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Mar 02 2020

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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.

Programs

  • Mathematica
    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]]]&]
Showing 1-8 of 8 results.

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