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.

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A344609 Numbers whose alternating sum of prime indices is >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 30, 31, 32, 36, 37, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 89, 92, 97, 98, 99, 100, 101, 102, 103, 105, 107
Offset: 1

Views

Author

Gus Wiseman, May 30 2021

Keywords

Comments

Also Heinz numbers of partitions whose reverse-alternating sum is >= 0. These are partitions whose conjugate parts are all even or whose length is odd.
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.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

Examples

			The sequence of terms together with their prime indices begins:
      1: {}            20: {1,1,3}         45: {2,2,3}
      2: {1}           23: {9}             47: {15}
      3: {2}           25: {3,3}           48: {1,1,1,1,2}
      4: {1,1}         27: {2,2,2}         49: {4,4}
      5: {3}           28: {1,1,4}         50: {1,3,3}
      7: {4}           29: {10}            52: {1,1,6}
      8: {1,1,1}       30: {1,2,3}         53: {16}
      9: {2,2}         31: {11}            59: {17}
     11: {5}           32: {1,1,1,1,1}     61: {18}
     12: {1,1,2}       36: {1,1,2,2}       63: {2,2,4}
     13: {6}           37: {12}            64: {1,1,1,1,1,1}
     16: {1,1,1,1}     41: {13}            66: {1,2,5}
     17: {7}           42: {1,2,4}         67: {19}
     18: {1,2,2}       43: {14}            68: {1,1,7}
     19: {8}           44: {1,1,5}         70: {1,3,4}
For example, the prime indices of 70 are {1,3,4} with alternating sum 1 - 3 + 4 = 2, so 70 is in the sequence. On the other hand, the prime indices of 24 are {1,1,1,2} with alternating sum 1 - 1 + 1 - 2 = -1, so 24 is not in the sequence.

Crossrefs

The opposite (nonpositive) version is A028260, counted by A027187.
The strict case (n > 0) is counted by A067659, odd bisection A344650.
Permutations of prime indices of these terms are counted by A116406.
Complement of A119899, Heinz numbers of the partitions counted by A344608.
Positions of nonnegative terms in A316524 or A344617.
Heinz numbers of the partitions counted by A344607.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A000070 counts partitions with alternating sum 1.
A000097 counts partitions with alternating sum 2.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum.
A120452 counts partitions with reverse-alternating sum 2.
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A335433/A335448 rank separable/inseparable partitions.
A344604 counts wiggly compositions with twins.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344612 counts partitions by sum and reverse-alternating sum.
A344618 gives reverse-alternating sums of standard compositions.
Cf. A001222, A001250, A003242, A005649, A026424, A071321/A071322, A124754, A239829, A343938, A344611, A344651, A344653/A344742, A344739.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[100],ats[primeMS[#]]>=0&]

A345918 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum > 0.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 11, 12, 14, 16, 19, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 35, 37, 38, 40, 42, 44, 47, 48, 51, 52, 54, 56, 59, 60, 62, 64, 67, 69, 70, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 87, 88, 91, 92, 93, 94, 96, 99, 100, 101, 102, 104, 106, 107, 108
Offset: 1

Views

Author

Gus Wiseman, Jul 09 2021

Keywords

Comments

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
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.

Examples

			The initial terms and the corresponding compositions:
     1: (1)        26: (1,2,2)        52: (1,2,3)
     2: (2)        27: (1,2,1,1)      54: (1,2,1,2)
     4: (3)        28: (1,1,3)        56: (1,1,4)
     6: (1,2)      30: (1,1,1,2)      59: (1,1,2,1,1)
     7: (1,1,1)    31: (1,1,1,1,1)    60: (1,1,1,3)
     8: (4)        32: (6)            62: (1,1,1,1,2)
    11: (2,1,1)    35: (4,1,1)        64: (7)
    12: (1,3)      37: (3,2,1)        67: (5,1,1)
    14: (1,1,2)    38: (3,1,2)        69: (4,2,1)
    16: (5)        40: (2,4)          70: (4,1,2)
    19: (3,1,1)    42: (2,2,2)        72: (3,4)
    20: (2,3)      44: (2,1,3)        73: (3,3,1)
    21: (2,2,1)    47: (2,1,1,1,1)    74: (3,2,2)
    22: (2,1,2)    48: (1,5)          76: (3,1,3)
    24: (1,4)      51: (1,3,1,1)      79: (3,1,1,1,1)

Crossrefs

The version for prime indices is A000037.
The version for Heinz numbers of partitions is A026424, counted by A027193.
These compositions are counted by A027306.
These are the positions of terms > 0 in A344618.
The weak (k >= 0) version is A345914.
The version for unreversed alternating sum is A345917.
The opposite (k < 0) version is A345920.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0: counted by A088218, ranked by A344619/A344619.
- k = 1: counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2: counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0: counted by A027306, ranked by A345917/A345918.
- k < 0: counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd: counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000290, A000346, A008549, A025047, A027187, A028260, A034871, A114121, A163493, A344607, A344608, A344650, A344743, A345908.

Programs

  • Mathematica
    stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,100],sats[stc[#]]>0&]

A345958 Numbers whose prime indices have reverse-alternating sum 1.

Original entry on oeis.org

2, 6, 8, 15, 18, 24, 32, 35, 50, 54, 60, 72, 77, 96, 98, 128, 135, 140, 143, 150, 162, 200, 216, 221, 240, 242, 288, 294, 308, 315, 323, 338, 375, 384, 392, 437, 450, 486, 512, 540, 560, 572, 578, 600, 648, 667, 693, 722, 726, 735, 800, 864, 875, 882, 884, 899
Offset: 1

Views

Author

Gus Wiseman, Jul 11 2021

Keywords

Comments

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.
The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. Of course, the reverse-alternating sum of prime indices is also the alternating sum of reversed prime indices.
Also numbers with exactly one odd conjugate prime index. Conjugate prime indices are listed by A321650, ranked by A122111.

Examples

			The initial terms and their prime indices:
   2: {1}
   6: {1,2}
   8: {1,1,1}
  15: {2,3}
  18: {1,2,2}
  24: {1,1,1,2}
  32: {1,1,1,1,1}
  35: {3,4}
  50: {1,3,3}
  54: {1,2,2,2}
  60: {1,1,2,3}
  72: {1,1,1,2,2}
  77: {4,5}
  96: {1,1,1,1,1,2}
  98: {1,4,4}

Crossrefs

The k > 0 version is A000037.
These multisets are counted by A000070.
The k = 0 version is A000290, counted by A000041.
The version for unreversed-alternating sum is A001105.
These partitions are counted by A035363.
These are the positions of 1's in A344616.
The k = 2 version is A345961, counted by A120452.
A000984/A345909/A345911 count/rank compositions with alternating sum 1.
A001791/A345910/A345912 count/rank compositions with alternating sum -1.
A088218 counts compositions with alternating sum 0, ranked by A344619.
A025047 counts wiggly compositions.
A027187 counts partitions with reverse-alternating sum <= 0.
A056239 adds up prime indices, row sums of A112798.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices.
A325534 and A325535 count separable and inseparable partitions.
A344606 counts alternating permutations of prime indices.
A344607 counts partitions with reverse-alternating sum >= 0.
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000097, A027193, A034871, A239830, A341446, A344650, A344651, A344743, A345917, A345918, A345920.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[100],sats[primeMS[#]]==1&]

A345914 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum >= 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 67, 69, 70, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 87, 88
Offset: 1

Views

Author

Gus Wiseman, Jul 04 2021

Keywords

Comments

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
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.

Examples

			The sequence of terms together with the corresponding compositions begins:
     0: ()           19: (3,1,1)        40: (2,4)
     1: (1)          20: (2,3)          41: (2,3,1)
     2: (2)          21: (2,2,1)        42: (2,2,2)
     3: (1,1)        22: (2,1,2)        43: (2,2,1,1)
     4: (3)          24: (1,4)          44: (2,1,3)
     6: (1,2)        26: (1,2,2)        46: (2,1,1,2)
     7: (1,1,1)      27: (1,2,1,1)      47: (2,1,1,1,1)
     8: (4)          28: (1,1,3)        48: (1,5)
    10: (2,2)        30: (1,1,1,2)      50: (1,3,2)
    11: (2,1,1)      31: (1,1,1,1,1)    51: (1,3,1,1)
    12: (1,3)        32: (6)            52: (1,2,3)
    13: (1,2,1)      35: (4,1,1)        53: (1,2,2,1)
    14: (1,1,2)      36: (3,3)          54: (1,2,1,2)
    15: (1,1,1,1)    37: (3,2,1)        55: (1,2,1,1,1)
    16: (5)          38: (3,1,2)        56: (1,1,4)

Crossrefs

The version for prime indices is A000027, counted by A000041.
These compositions are counted by A116406.
The case of non-Heinz numbers of partitions is A119899, counted by A344608.
The version for Heinz numbers of partitions is A344609, counted by A344607.
These are the positions of terms >= 0 in A344618.
The version for unreversed alternating sum is A345913.
The opposite (k <= 0) version is A345916.
The strict (k > 0) case is A345918.
The complement is A345920, counted by A294175.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0: counted by A088218, ranked by A344619/A344619.
- k = 1: counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2: counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0: counted by A027306, ranked by A345917/A345918.
- k < 0: counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd: counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000346, A008549, A025047, A027187, A032443, A034871, A114121, A120452, A163493, A238279, A344650, A344743.

Programs

  • Mathematica
    stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,100],sats[stc[#]]>=0&]

A344739 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with reverse-alternating sum k, with k ranging from -n to n in steps of 2.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 0, 1, 0, 1, 1, 1, 2, 0, 1, 2, 1, 0, 1, 0, 1, 1, 1, 2, 1, 0, 2, 2, 1, 0, 1
Offset: 0

Views

Author

Gus Wiseman, Jun 05 2021

Keywords

Comments

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. So T(n,k) is the number of strict integer partitions of n such that k is equal to (-1)^(m-1) times the number of odd conjugate parts.
By conjugation, T(n,k) is also equal to the number of integer partitions of n covering an initial interval of positive integers such that k is equal to (-1)^(r-1) times the number of odd parts, where r is the greatest part.
Also the number of reversed strict integer partitions of n with alternating sum k.

Examples

			Triangle begins:
                                      1
                                    0   1
                                  0   0   1
                                0   1   0   1
                              0   1   0   0   1
                            0   1   1   0   0   1
                          0   1   1   0   1   0   1
                        0   1   1   1   0   1   0   1
                      0   1   1   1   0   1   1   0   1
                    0   1   1   1   1   0   2   1   0   1
                  0   1   1   1   2   0   1   2   1   0   1
                0   1   1   1   2   1   0   2   2   1   0   1
              0   1   1   1   2   2   0   1   3   2   1   0   1
            0   1   1   1   2   3   1   0   2   3   2   1   0   1
          0   1   1   1   2   3   3   0   1   3   3   2   1   0   1
        0   1   1   1   2   3   4   1   0   3   4   3   2   1   0   1
      0   1   1   1   2   3   5   3   0   1   4   4   3   2   1   0   1
    0   1   1   1   2   3   5   5   1   0   3   5   4   3   2   1   0   1
  0   1   1   1   2   3   5   6   4   0   1   5   6   4   3   2   1   0   1
For example, the partitions counted by row n = 15 are (empty columns shown as dots, A...F = 10..15):
  .  E1  D2  C3  B4    A5    96    87  .  762    654  843  A32  C21  .  F
                 9321  7431  6432         861    753  942  B31
                       8421  6531         54321  852  A41
                             7521                951

Crossrefs

Row sums are A000009.
The non-reverse version is A152146 interleaved with A152157.
The non-strict version is A344612.
The right halves of even-indexed rows are A344649.
The non-reverse non-strict version is the right half of A344651, which is A239830 interleaved with A239829.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A124754 lists alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344610 counts partitions of n by positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A000070, A003242, A006330, A027187, A103919, A114121, A343941, A344607, A344608, A344650, A344654.

Programs

  • Mathematica
    sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&sats[#]==k&]],{n,0,12},{k,-n,n,2}]

A344741 Number of integer partitions of 2n with reverse-alternating sum -2.

Original entry on oeis.org

0, 0, 1, 2, 4, 8, 14, 24, 39, 62, 95, 144, 212, 309, 442, 626, 873, 1209, 1653, 2245, 3019, 4035, 5348, 7051, 9229, 12022, 15565, 20063, 25722, 32847, 41746, 52862, 66657, 83768, 104873, 130889, 162797, 201902, 249620, 307789, 378428, 464122, 567721, 692828, 843448
Offset: 0

Views

Author

Gus Wiseman, Jun 08 2021

Keywords

Comments

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(r-1) times the number of odd parts, where r is the greatest part, so a(n) is the number of integer partitions of 2n with exactly two odd parts, neither of which is the greatest.
Also the number of reversed integer partitions of 2n with alternating sum -2.

Examples

			The a(2) = 1 through a(6) = 14 partitions:
  (31)  (42)    (53)      (64)        (75)
        (3111)  (3221)    (3331)      (4332)
                (4211)    (4222)      (4431)
                (311111)  (4321)      (5322)
                          (5311)      (5421)
                          (322111)    (6411)
                          (421111)    (322221)
                          (31111111)  (333111)
                                      (422211)
                                      (432111)
                                      (531111)
                                      (32211111)
                                      (42111111)
                                      (3111111111)

Crossrefs

The version for -1 instead of -2 is A000070.
The non-reversed negative version is A000097.
The ordered version appears to be A001700.
The version for 1 instead of -2 is A035363.
The whole set of partitions of 2n is counted by A058696.
The strict case appears to be A065033.
The version for -1 instead of -2 is A306145.
The version for 2 instead of -2 is A344613.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A124754 gives alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A001250, A003242, A006330, A027187, A028260, A344604, A344607, A344608, A344650, A344651, A344654, A344739.

Programs

  • Mathematica
    sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],sats[#]==-2&]],{n,0,30,2}]
- or -
Table[Length[Select[IntegerPartitions[n],EvenQ[Max[#]]&&Count[#,_?OddQ]==2&]],{n,0,30,2}]

Extensions

More terms from Bert Dobbelaere, Jun 12 2021

A152146 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) = number of partitions of 2n into 2k odd parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 1, 0, 3, 3, 2, 1, 1, 0, 3, 5, 3, 2, 1, 1, 0, 4, 6, 5, 3, 2, 1, 1, 0, 4, 9, 7, 5, 3, 2, 1, 1, 0, 5, 11, 11, 7, 5, 3, 2, 1, 1, 0, 5, 15, 14, 11, 7, 5, 3, 2, 1, 1, 0, 6, 18, 20, 15, 11, 7, 5, 3, 2, 1, 1, 0, 6, 23, 26, 22, 15, 11, 7, 5, 3, 2, 1, 1
Offset: 0

Views

Author

R. J. Mathar, Sep 25 2009, indices corrected Jul 09 2012

Keywords

Comments

In both this and A152157, reading columns downwards "converges" to A000041.
Also the number of strict integer partitions of 2n with alternating sum 2k. Also the number of normal integer partitions of 2n of which 2k parts are odd, where a partition is normal if it covers an initial interval of positive integers. - Gus Wiseman, Jun 20 2021

Examples

			Triangle begins:
  1
  0  1
  0  1  1
  0  2  1   1
  0  2  2   1   1
  0  3  3   2   1   1
  0  3  5   3   2   1   1
  0  4  6   5   3   2   1  1
  0  4  9   7   5   3   2  1  1
  0  5 11  11   7   5   3  2  1  1
  0  5 15  14  11   7   5  3  2  1  1
  0  6 18  20  15  11   7  5  3  2  1  1
  0  6 23  26  22  15  11  7  5  3  2  1  1
  0  7 27  35  29  22  15 11  7  5  3  2  1  1
  0  7 34  44  40  30  22 15 11  7  5  3  2  1 1
  0  8 39  58  52  42  30 22 15 11  7  5  3  2 1 1
  0  8 47  71  70  55  42 30 22 15 11  7  5  3 2 1 1
  0  9 54  90  89  75  56 42 30 22 15 11  7  5 3 2 1 1
  0  9 64 110 116  97  77 56 42 30 22 15 11  7 5 3 2 1 1
  0 10 72 136 146 128 100 77 56 42 30 22 15 11 7 5 3 2 1 1
From _Gus Wiseman_, Jun 20 2021: (Start)
For example, row n = 6 counts the following partitions (B = 11):
  (75)  (3333)  (333111)  (33111111)  (3111111111)  (111111111111)
  (93)  (5331)  (531111)  (51111111)
  (B1)  (5511)  (711111)
        (7311)
        (9111)
The corresponding strict partitions are:
  (7,5)      (8,4)      (9,3)    (10,2)   (11,1)  (12)
  (6,5,1)    (5,4,3)    (7,3,2)  (9,2,1)
  (5,4,2,1)  (6,4,2)    (8,3,1)
             (7,4,1)
             (6,3,2,1)
The corresponding normal partitions are:
  43221    33321     3321111    321111111   21111111111  111111111111
  322221   332211    32211111   2211111111
  2222211  432111    222111111
           3222111
           22221111
(End)

Links

Crossrefs

Cf. A035294 (row sums), A107379, A152140, A152157.
Column k = 1 is A004526.
Column k = 2-8 is A026810 - A026816.
The non-strict version is A239830.
The reverse non-strict version is A344610.
The reverse version is A344649
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A067659 counts strict partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A124754 gives alternating sum of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A006330, A027187, A116406, A120452, A239829, A343941, A344607, A344608, A344609, A344650, A344651, A344739, A344741.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-2)+`if`(i>n, 0, expand(sqrt(x)*b(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(2*n, 2*n-1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Jun 21 2021
  • Mathematica
    ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&ats[#]==k&]],{n,0,30,2},{k,0,n,2}] (* Gus Wiseman, Jun 20 2021 *)

Formula

T(n,k) = A152140(2n,2k).

A344649 Triangle read by rows where T(n,k) is the number of strict integer partitions of 2n with reverse-alternating sum 2k.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 1, 3, 3, 2, 1, 0, 1, 0, 1, 4, 4, 3, 2, 1, 0, 1, 0, 1, 5, 6, 4, 3, 2, 1, 0, 1, 0, 1, 7, 7, 6, 4, 3, 2, 1, 0, 1, 0, 1, 8, 10, 8, 6, 4, 3, 2, 1, 0, 1
Offset: 0

Views

Author

Gus Wiseman, Jun 05 2021

Keywords

Comments

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. So T(n,k) is the number of strict integer partitions of 2n into an odd number of parts whose conjugate has exactly 2k odd parts.
Also the number of reversed strict integer partitions of 2n with alternating sum 2k.

Examples

			Triangle begins:
   1
   0   1
   0   0   1
   0   1   0   1
   0   1   1   0   1
   0   1   2   1   0   1
   0   1   3   2   1   0   1
   0   1   3   3   2   1   0   1
   0   1   4   4   3   2   1   0   1
   0   1   5   6   4   3   2   1   0   1
   0   1   7   7   6   4   3   2   1   0   1
   0   1   8  10   8   6   4   3   2   1   0   1
   0   1  10  13  12   8   6   4   3   2   1   0   1
   0   1  11  18  15  12   8   6   4   3   2   1   0   1
   0   1  14  22  21  16  12   8   6   4   3   2   1   0   1
   0   1  15  29  27  23  16  12   8   6   4   3   2   1   0   1
Row n = 8 counts the following partitions (empty columns indicated by dots):
  .  (8,7,1)  (7,6,3)      (7,5,4)   (9,4,3)   (11,3,2)  (13,2,1)  .  (16)
              (8,6,2)      (8,5,3)   (10,4,2)  (12,3,1)
              (9,6,1)      (9,5,2)   (11,4,1)
              (6,4,3,2,1)  (10,5,1)
Row n = 9 counts the following partitions (empty columns indicated by dots, A..I = 10..18):
  .  981   873     765     954   B43   D32   F21   .  I
           972     864     A53   C42   E31
           A71     963     B52   D41
           65421   A62     C51
           75321   B61
                   84321

Crossrefs

The non-reversed version is A152146.
The non-reversed non-strict version is A239830.
Column k = 2 is A343941.
The non-strict version is A344610.
Row sums are A344650.
Right half of even-indexed rows of A344739.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A067659 counts strict partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with reverse-alternating sum 2.
A124754 gives alternating sum of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344604 counts wiggly compositions with twins.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A344741 counts partitions of 2n with reverse-alternating sum -2.
Cf. A000070, A000097, A003242, A006330, A027187, A114121, A116406, A239829, A344607, A344608, A344651, A344654.

Programs

  • Mathematica
    sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&sats[#]==k&]],{n,0,30,2},{k,0,n,2}]

A344743 Number of integer partitions of 2n with reverse-alternating sum < 0.

Original entry on oeis.org

0, 0, 1, 3, 7, 15, 29, 54, 96, 165, 275, 449, 716, 1123, 1732, 2635, 3955, 5871, 8620, 12536, 18065, 25821, 36617, 51560, 72105, 100204, 138417, 190134, 259772, 353134, 477734, 643354, 862604, 1151773, 1531738, 2029305, 2678650, 3523378, 4618835, 6035240, 7861292
Offset: 0

Views

Author

Gus Wiseman, Jun 09 2021

Keywords

Comments

Conjecture: a(n) >= A236914.
The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. So a(n) is the number of even-length partitions of 2n with at least one odd conjugate part. By conjugation, this is also the number of partitions of 2n with greatest part even and at least one odd part.
The alternating sum of a partition is never < 0, so the non-reverse version is A000004.

Examples

			The a(2) = 1 through a(5) = 15 partitions:
  (31)  (42)    (53)      (64)
        (51)    (62)      (73)
        (3111)  (71)      (82)
                (3221)    (91)
                (4211)    (3331)
                (5111)    (4222)
                (311111)  (4321)
                          (5221)
                          (5311)
                          (6211)
                          (7111)
                          (322111)
                          (421111)
                          (511111)
                          (31111111)

Crossrefs

The ordered version (compositions not partitions) appears to be A008549.
The Heinz numbers are A119899 /\ A300061.
Even bisection of A344608.
The complementary partitions of 2n are counted by A344611.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A001523 counts unimodal compositions (partial sums: A174439).
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
A124754 gives alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000070, A000097, A006330, A027187, A239830, A343941, A344604, A344607, A344650, A344651, A344654.

Programs

  • Mathematica
    sats[y_] := Sum[(-1)^(i - Length[y])*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n],sats[#]<0&]],{n,0,30,2}]

Formula

a(n) = A058696(n) - A344611(n).
a(n) = sum of left half of even-indexed rows of A344612.

Extensions

More terms from Bert Dobbelaere, Jun 12 2021

A343941 Number of strict integer partitions of 2n with reverse-alternating sum 4.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 3, 4, 5, 7, 8, 10, 11, 14, 15, 18, 20, 23, 25, 29, 31, 35, 38, 42, 45, 50, 53, 58, 62, 67, 71, 77, 81, 87, 92, 98, 103, 110, 115, 122, 128, 135, 141, 149, 155, 163, 170, 178, 185, 194, 201, 210, 218, 227, 235, 245, 253, 263, 272, 282, 291, 302
Offset: 0

Views

Author

Gus Wiseman, Jun 09 2021

Keywords

Comments

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts, so a(n) is the number of strict odd-length integer partitions of 2n whose conjugate has exactly 4 odd parts (first example). By conjugation, this is also the number partitions of 2n covering an initial interval and containing exactly four odd parts, one of which is the greatest (second example).

Examples

			The a(2) = 1 through a(12) = 10 strict partitions (empty column indicated by dot, A..D = 10..13):
  4   .  521   532   543   653   763     873     983     A93     BA3
               631   642   752   862     972     A82     B92     CA2
                     741   851   961     A71     B81     C91     DA1
                                 64321   65421   65432   76432   76542
                                         75321   75431   76531   86541
                                                 76421   86431   87432
                                                 86321   87421   87531
                                                         97321   97431
                                                                 98421
                                                                 A8321
The a(2) = 1 through a(8) = 5 partitions covering an initial interval:
  1111  .  32111   33211    33321     333221     543211      543321
                   322111   332211    3322211    3332221     5432211
                            3222111   32222111   33222211    33322221
                                                 322222111   332222211
                                                             3222222111

Crossrefs

The non-reverse non-strict version is A000710.
The non-reverse version is A026810.
The non-strict version is column k = 2 of A344610.
This is column k = 2 of A344649.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
A124754 gives alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A000070, A000097, A003242, A006330, A027187, A119899, A152146, A239830, A325535, A344604, A344607, A344608, A344650, A344739.

Programs

  • Mathematica
    sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&sats[#]==4&]],{n,0,30,2}]

Extensions

More terms from Bert Dobbelaere, Jun 12 2021
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