A035594 -id:A035594 - cp's OEIS frontend

A357639 Number of reversed integer partitions of 2n whose half-alternating sum is 0.

1, 0, 2, 1, 6, 4, 15, 13, 37, 37, 86, 94, 194, 223, 416, 497, 867, 1056, 1746, 2159, 3424, 4272, 6546, 8215, 12248, 15418, 22449, 28311, 40415, 50985, 71543, 90222, 124730, 157132, 214392, 269696, 363733, 456739, 609611, 763969, 1010203, 1263248, 1656335, 2066552, 2688866

Offset: 0

Gus Wiseman, Oct 11 2022

nonn

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...

			The a(0) = 1 through a(6) = 15 reversed partitions:
  ()  .  (112)   (123)  (134)       (145)      (156)
         (1111)         (224)       (235)      (246)
                        (2222)      (11233)    (336)
                        (11222)     (1111123)  (3333)
                        (1111112)              (11244)
                        (11111111)             (11334)
                                               (12333)
                                               (1111134)
                                               (1111224)
                                               (1112223)
                                               (1122222)
                                               (11112222)
                                               (111111222)
                                               (11111111112)
                                               (111111111111)

Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 51 terms from Lucas A. Brown)
Lucas A. Brown, A357639.py.

The non-reverse version is A035363/A035444.
The non-reverse skew version appears to be A035544/A035594.
These partitions are ranked by A357631, skew A357632.
The skew-alternating version is A357640.
This is the central column of A357704.
A000041 counts integer partitions (also reversed integer partitions).
A316524 gives alternating sum of prime indices, reverse A344616.
A344651 counts alternating sum of partitions by length, ordered A097805.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357629 gives half-alternating sum of prime indices, skew A357630.
A357633 gives half-alternating sum of Heinz partition, skew  A357634.
A357637 counts partitions by half-alternating sum, skew A357637.
Cf. A029862, A053251, A357189, A357487, A357488, A357631, A357634, A357636, A357639, A357641, A357645.

halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Table[Length[Select[IntegerPartitions[2n],halfats[Reverse[#]]==0&]],{n,0,15}]

a(31) onwards from Lucas A. Brown, Oct 19 2022

A357638 Triangle read by rows where T(n,k) is the number of integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 3, 1, 1, 0, 0, 1, 4, 1, 1, 0, 0, 1, 4, 4, 1, 1, 0, 0, 0, 4, 5, 4, 1, 1, 0, 0, 0, 1, 10, 5, 4, 1, 1, 0, 0, 0, 1, 5, 13, 5, 4, 1, 1, 0, 0, 0, 0, 4, 13, 14, 5, 4, 1, 1, 0, 0, 0, 0, 1, 13, 17, 14, 5, 4, 1, 1

Offset: 0

Gus Wiseman, Oct 10 2022

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  0  3  1  1
  0  0  1  4  1  1
  0  0  1  4  4  1  1
  0  0  0  4  5  4  1  1
  0  0  0  1 10  5  4  1  1
  0  0  0  1  5 13  5  4  1  1
  0  0  0  0  4 13 14  5  4  1  1
  0  0  0  0  1 13 17 14  5  4  1  1
  0  0  0  0  1  5 28 18 14  5  4  1  1
Row n = 7 counts the following partitions:
  .  .  .  (322)      (43)      (52)     (61)  (7)
           (331)      (421)     (511)
           (2221)     (3211)    (4111)
           (1111111)  (22111)   (31111)
                      (211111)

Row sums are A000041.
Number of nonzero entries in row n appears to be A004396(n+1).
First nonzero entry of each row appears to converge to A146325.
The central column is A035544, half A035363.
Column sums appear to be A298311.
For original alternating sum we have A344651, ordered A097805.
The half-alternating version is A357637.
The ordered version (compositions) is A357646, half A357645.
The reverse version is A357705, half A357704.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357629 gives half-alternating sum of prime indices, skew A357630.
A357633 gives half-alternating sum of Heinz partition, skew  A357634.
Cf. A035594, A053251, A357136, A357189, A357486, A357487, A357488, A357624, A357631, A357632, A357636.

skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Table[Length[Select[IntegerPartitions[n],skats[#]==k&]],{n,0,12},{k,-n,n,2}]

Conjecture: The columns are palindromes with sums A298311.

A357640 Number of reversed integer partitions of 2n whose skew-alternating sum is 0.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 16, 24, 40, 59, 93, 136, 208, 299, 445, 632, 921, 1292, 1848, 2563, 3610, 4954, 6881, 9353, 12835, 17290, 23469, 31357, 42150, 55889, 74463, 98038, 129573, 169476, 222339, 289029, 376618, 486773, 630313, 810285, 1043123, 1334174

Offset: 0

Gus Wiseman, Oct 11 2022

nonn

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ...

			The a(0) = 1 through a(5) = 9 partitions:
  ()  (11)  (22)    (33)      (44)        (55)
            (1111)  (2211)    (2222)      (3322)
                    (111111)  (3221)      (4321)
                              (3311)      (4411)
                              (221111)    (222211)
                              (11111111)  (322111)
                                          (331111)
                                          (22111111)
                                          (1111111111)

Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 51 terms from Lucas A. Brown)
Lucas A. Brown, A357640.py.

The non-reverse half-alternating version is A035363/A035444.
The non-reverse version appears to be A035544/A035594.
These partitions are ranked by A357632, half A357631.
The half-alternating version is A357639.
A000041 counts integer partitions (also reversed integer partitions).
A316524 gives alternating sum of prime indices, reverse A344616.
A344651 counts alternating sum of partitions by length, ordered A097805.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357629 gives half-alternating sum of prime indices, skew A357630.
A357633 gives half-alternating sum of Heinz partition, skew  A357634.
A357637 counts partitions by half-alternating sum, skew A357638.
Cf. A029862, A053251, A357136, A357189, A357487, A357488, A357636, A357641, A357645, A357704.

skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Table[Length[Select[IntegerPartitions[2n],skats[Reverse[#]]==0&]],{n,0,15}]

a(31) onwards from Lucas A. Brown, Oct 19 2022

A357636 Numbers k such that the skew-alternating sum of the partition having Heinz number k is 0.

Original entry on oeis.org

1, 4, 9, 12, 16, 25, 30, 36, 49, 63, 64, 70, 81, 90, 100, 108, 121, 144, 154, 165, 169, 192, 196, 210, 225, 256, 273, 286, 289, 300, 324, 325, 360, 361, 400, 441, 442, 462, 480, 484, 525, 529, 550, 561, 576, 588, 595, 625, 646, 676, 700, 729, 741, 750, 784

Offset: 1

Gus Wiseman, Oct 09 2022

nonn

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
    1: {}
    4: {1,1}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   25: {3,3}
   30: {1,2,3}
   36: {1,1,2,2}
   49: {4,4}
   63: {2,2,4}
   64: {1,1,1,1,1,1}
   70: {1,3,4}
   81: {2,2,2,2}
   90: {1,2,2,3}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  121: {5,5}
  144: {1,1,1,1,2,2}

The version for original alternating sum is A000290.
The half-alternating form is A000583, non-reverse A357631.
The version for standard compositions is A357628, non-reverse A357627.
The non-reverse version is A357632.
Positions of zeros in A357634, non-reverse A357630.
These partitions are counted by A357640, half A357639.
A056239 adds up prime indices, row sums of A112798.
A316524 gives alternating sum of prime indices, reverse A344616.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, even A357642.
Cf. A003963, A035594, A053251, A055932, A357189, A357485-A357488, A357621-A357626, A357629, A357637, A357638.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Select[Range[1000],skats[Reverse[primeMS[#]]]==0&]

A035544 Number of partitions of n with equal number of parts congruent to each of 1 and 3 (mod 4).

Original entry on oeis.org

1, 0, 1, 0, 3, 0, 4, 0, 10, 0, 13, 0, 28, 0, 37, 0, 72, 0, 96, 0, 172, 0, 230, 0, 391, 0, 522, 0, 846, 0, 1129, 0, 1766, 0, 2348, 0, 3564, 0, 4722, 0, 6992, 0, 9226, 0, 13371, 0, 17568, 0, 25006, 0, 32708, 0, 45817, 0, 59668, 0, 82430, 0, 106874, 0, 145830, 0, 188260, 0

Offset: 0

Olivier Gérard

nonn

From Gus Wiseman, Oct 12 2022: (Start)
Also the number of integer partitions of n whose skew-alternating sum is 0, where we define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ... These are the conjugates of the partitions described in the name. For example, the a(0) = 1 through a(8) = 10 partitions are:
  ()  .  (11)  .  (22)    .  (33)      .  (44)
                  (211)      (321)        (422)
                  (1111)     (2211)       (431)
                             (111111)     (2222)
                                          (3221)
                                          (3311)
                                          (22211)
                                          (221111)
                                          (2111111)
                                          (11111111)
The ordered version (compositions) is A138364
These partitions are ranked by A357636, reverse A357632.
The reverse version is A357640 (aerated).
Cf. A357623, A357629, A357630, A357634, A357646, A357705.
(End)

			From _Gus Wiseman_, Oct 12 2022: (Start)
The a(0) = 1 through a(8) = 10 partitions:
  ()  .  (2)  .  (4)   .  (6)    .  (8)
                 (22)     (42)      (44)
                 (31)     (222)     (53)
                          (321)     (62)
                                    (71)
                                    (422)
                                    (431)
                                    (2222)
                                    (3221)
                                    (3311)
(End)

The case with at least one odd part is A035550.
Removing zeros gives A035594.
Central column k=0 of A357638.
These partitions are ranked by A357707.
A000041 counts integer partitions.
A344651 counts partitions by alternating sum, ordered A097805.
Cf. A035363, A053251, A298311, A357189, A357487, A357488.

skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Table[Length[Select[IntegerPartitions[n],skats[#]==0&]],{n,0,30}] (* Gus Wiseman,Oct 12 2022 *)

More terms from David W. Wilson

A357705 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 2, 0, 1, 0, 3, 1, 2, 0, 1, 0, 3, 2, 3, 2, 0, 1, 0, 4, 2, 4, 1, 3, 0, 1, 0, 4, 3, 3, 6, 2, 3, 0, 1, 0, 5, 3, 5, 3, 7, 2, 4, 0, 1, 0, 5, 4, 5, 4, 9, 7, 3, 4, 0, 1, 0, 6, 4, 7, 3, 12, 5, 10, 3, 5, 0, 1

Offset: 0

Gus Wiseman, Oct 10 2022

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ...

			Triangle begins:
  1
  0  1
  0  1  1
  0  2  0  1
  0  2  2  0  1
  0  3  1  2  0  1
  0  3  2  3  2  0  1
  0  4  2  4  1  3  0  1
  0  4  3  3  6  2  3  0  1
  0  5  3  5  3  7  2  4  0  1
  0  5  4  5  4  9  7  3  4  0  1
  0  6  4  7  3 12  5 10  3  5  0  1
  0  6  5  7  5 10 16  7 11  4  5  0  1
  0  7  5  9  5 14 11 18  7 14  4  6  0  1
Row n = 7 counts the following reversed partitions:
  .  (16)   (25)   (34)       (1123)  (1114)   .  (7)
     (115)  (223)  (1222)             (11113)
     (124)         (111112)           (11122)
     (133)         (1111111)

Row sums are A000041.
First nonzero entry of each row is A004526.
The central column is A357640, half A357639.
For original alternating sum we have A344651, ordered A097805.
The half-alternating version is A357704.
The ordered non-reverse version (compositions) is A357646, half A357645.
The non-reverse version is A357638, half A357637.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357629 gives half-alternating sum of prime indices, skew A357630.
A357633 gives half-alternating sum of Heinz partition, skew  A357634.
Cf. A035363, A035594, A053251, A298311, A357136, A357189, A357487, A357488, A357624, A357631, A357632, A357636.

skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Table[Length[Select[Reverse/@IntegerPartitions[n],skats[#]==k&]],{n,0,11},{k,-n,n,2}]

A357706 Numbers k such that the k-th composition in standard order has half-alternating sum and skew-alternating sum both 0.

Original entry on oeis.org

0, 15, 45, 54, 59, 153, 170, 179, 204, 213, 230, 235, 247, 255, 561, 594, 611, 660, 677, 710, 715, 727, 735, 750, 765, 792, 809, 842, 851, 871, 879, 894, 908, 917, 934, 939, 951, 959, 973, 982, 987, 1005, 1014, 1019

Offset: 1

Gus Wiseman, Oct 13 2022

nonn

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ..., and the skew-alternating sum to be A - B - C + D + E - F - G + ...

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.

For partitions and half only (or both): A000583, counted by A035363.
These compositions are counted by A228248.
For half-alternating only: A357625, reverse A357626, counted by A357641.
For skew-alternating only: A357627, reverse A357628, counted by A001700.
For reversed partitions and half only: A357631, counted by A357639.
For reversed partitions and skew only A357632, counted by A357640.
For partitions and skew only: A357636, counted by A035594.
Cf. A001511, A004006, A088218, A357136, A357182, A357642.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Select[Range[0,1000],halfats[stc[#]]==0&&skats[stc[#]]==0&]

Intersection of A357625 and A357627.

A357707 Numbers whose prime indices have equal number of parts congruent to each of 1 and 3 (mod 4).

Original entry on oeis.org

1, 3, 7, 9, 10, 13, 19, 21, 27, 29, 30, 34, 37, 39, 43, 49, 53, 55, 57, 61, 62, 63, 70, 71, 79, 81, 87, 89, 90, 91, 94, 100, 101, 102, 107, 111, 113, 115, 117, 129, 130, 131, 133, 134, 139, 147, 151, 159, 163, 165, 166, 169, 171, 173, 181, 183, 186, 187, 189

Offset: 1

Gus Wiseman, Oct 12 2022

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.

			The terms together with their prime indices begin:
     1: {}
     3: {2}
     7: {4}
     9: {2,2}
    10: {1,3}
    13: {6}
    19: {8}
    21: {2,4}
    27: {2,2,2}
    29: {10}
    30: {1,2,3}

These partitions are counted by A035544.
Includes A066207 = products of primes of even index.
The conjugate partitions are ranked by A357636, reverse A357632.
The conjugate reverse version is A357640 (aerated).
A056239 adds up prime indices, row sums of A112798.
A316524 gives alternating sum of prime indices, reverse A344616.
A344651 counts partitions by alternating sum, ordered A097805.
A357705 counts reversed partitions by skew-alternating sum, half A357704.
Cf. A035363, A035550, A035594, A053251, A298311, A357486, A357623, A357638.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[primeMS[#],?(Mod[#,4]==1&)]==Count[primeMS[#],?(Mod[#,4]==3&)]&]

cp's OEIS Frontend

A357639 Number of reversed integer partitions of 2n whose half-alternating sum is 0.

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A357638 Triangle read by rows where T(n,k) is the number of integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.

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A357640 Number of reversed integer partitions of 2n whose skew-alternating sum is 0.

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A357636 Numbers k such that the skew-alternating sum of the partition having Heinz number k is 0.

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A035544 Number of partitions of n with equal number of parts congruent to each of 1 and 3 (mod 4).

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A357705 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.

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A357706 Numbers k such that the k-th composition in standard order has half-alternating sum and skew-alternating sum both 0.

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A357707 Numbers whose prime indices have equal number of parts congruent to each of 1 and 3 (mod 4).

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