A357633 -id:A357633 - cp's OEIS frontend

A357626 Numbers k such that the reversed k-th composition in standard order has half-alternating sum 0.

0, 11, 15, 37, 38, 45, 46, 53, 54, 55, 59, 137, 138, 140, 153, 154, 156, 167, 169, 170, 171, 172, 179, 191, 201, 202, 204, 205, 206, 213, 214, 229, 230, 231, 235, 243, 247, 251, 255, 529, 530, 532, 536, 561, 562, 564, 568, 583, 587, 593, 594, 595, 596, 600

Offset: 1

Gus Wiseman, Oct 08 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 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:
    0: ()
   11: (2,1,1)
   15: (1,1,1,1)
   37: (3,2,1)
   38: (3,1,2)
   45: (2,1,2,1)
   46: (2,1,1,2)
   53: (1,2,2,1)
   54: (1,2,1,2)
   55: (1,2,1,1,1)
   59: (1,1,2,1,1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
The alternating form is A344619.
Positions of zeros in A357622.
The non-reverse version is A357625.
The skew-alternating form is A357628, reverse A357627.
The version for prime indices is A357631.
The version for Heinz numbers of partitions is A357635.
A124754 gives alternating sum of standard compositions, reverse A344618.
A357637 counts partitions by half-alternating sum, skew A357638.
A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
Cf. A001511, A053251, A357136, A357182, A357183, A357184, A357185, A357621, A357623, A357629, A357633.

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]}];
Select[Range[0,100],halfats[Reverse[stc[#]]]==0&]

A357630 Skew-alternating sum of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 0, 3, -1, 4, -1, 0, -2, 5, -2, 6, -3, -1, 0, 7, -3, 8, -3, -2, -4, 9, 1, 0, -5, -2, -4, 10, -4, 11, 1, -3, -6, -1, 0, 12, -7, -4, 2, 13, -5, 14, -5, -3, -8, 15, 2, 0, -5, -5, -6, 16, -1, -2, 3, -6, -9, 17, 1, 18, -10, -4, 0, -3, -6, 19, -7, -7, -6, 20

Offset: 1

Gus Wiseman, Oct 09 2022

sign

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 + ....

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 prime indices of 525 are {2,3,3,4} so a(525) = 2 - 3 - 3 + 4 = 0.

The original alternating sum is A316524, reverse A344616.
The reverse version is A357634.
The half-alternating form is A357629, reverse A357633.
Positions of zeros are A357632, reverse A357636.
The version for standard compositions is A357623, reverse A357624.
These partitions are counted by A357638, half A357637.
A056239 adds up prime indices, row sums of A112798.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357621, A357622, A357625, A357626, A357635, A357640.

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]}];
Table[skats[primeMS[n]],{n,30}]

A357634 Skew-alternating sum of the partition having Heinz number n.

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 4, -1, 0, 2, 5, 0, 6, 3, 1, 0, 7, -1, 8, 1, 2, 4, 9, 1, 0, 5, -2, 2, 10, 0, 11, 1, 3, 6, 1, 0, 12, 7, 4, 2, 13, 1, 14, 3, -1, 8, 15, 2, 0, -1, 5, 4, 16, -1, 2, 3, 6, 9, 17, 1, 18, 10, 0, 0, 3, 2, 19, 5, 7, 0, 20, 1, 21, 11, -2, 6, 1, 3, 22, 3

Offset: 1

Gus Wiseman, Oct 09 2022

sign

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 partition with Heinz number 525 is (4,3,3,2) so a(525) = 4 - 3 - 3 + 2 = 0.

The original alternating sum is A316524, reverse A344616.
The non-reverse version is A357630.
The half-alternating form is A357633, non-reverse A357629.
Positions of zeros are A357636, non-reverse A357632.
The version for standard compositions is A357624, non-reverse A357623.
These partitions are counted by A357638, half A357637.
A056239 adds up prime indices, row sums of A112798.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357621, A357622, A357625, A357626, A357635, A357640.

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]}];
Table[skats[Reverse[primeMS[n]]],{n,30}]

A357622 Half-alternating sum of the reversed n-th composition in standard order.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 1, 4, 4, 4, 0, 4, 2, 2, 0, 5, 5, 5, -1, 5, 1, 1, -1, 5, 3, 3, -1, 3, 1, 1, 1, 6, 6, 6, -2, 6, 0, 0, -2, 6, 2, 2, -2, 2, 0, 0, 2, 6, 4, 4, -2, 4, 0, 0, 0, 4, 2, 2, 0, 2, 2, 2, 2, 7, 7, 7, -3, 7, -1, -1, -3, 7, 1, 1, -3, 1, -1, -1, 3, 7, 3

Offset: 0

Gus Wiseman, Oct 08 2022

sign

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 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 357-th composition is (2,1,3,2,1) so a(357) = 1 + 2 - 3 - 1 + 2 = 1.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
This is the reverse version of A357621.
The skew-alternating form is A357624, non-reverse A357623.
Positions of zeros are A357626, reverse A357625.
The version for prime indices is A357629.
The version for Heinz numbers of partitions is A357633.
A124754 gives alternating sum of standard compositions, reverse A344618.
A357637 counts partitions by half-alternating sum, skew A357638.
A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
Cf. A001511, A053251, A357136, A357182, A357183, A357184, A357185, A357627, A357628, A357631, A357635.

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]}];
Table[halfats[Reverse[stc[n]]],{n,0,100}]

A357635 Numbers k such that the half-alternating sum of the partition having Heinz number k is 1.

Original entry on oeis.org

2, 8, 24, 32, 54, 128, 135, 162, 375, 384, 512, 648, 864, 875, 1250, 1715, 1944, 2048, 2160, 2592, 3773, 4374, 4802, 5000, 6000, 6144, 8192, 9317, 10368, 10935, 13122, 13824, 14000, 15000, 17303, 19208, 20000, 24167, 27440, 29282, 30375, 31104, 32768, 33750

Offset: 1

Gus Wiseman, Oct 28 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 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:
    2: {1}
    8: {1,1,1}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   54: {1,2,2,2}
  128: {1,1,1,1,1,1,1}
  135: {2,2,2,3}
  162: {1,2,2,2,2}
  375: {2,3,3,3}
  384: {1,1,1,1,1,1,1,2}
  512: {1,1,1,1,1,1,1,1,1}
  648: {1,1,1,2,2,2,2}
  864: {1,1,1,1,1,2,2,2}
  875: {3,3,3,4}

The version for k = 0 is A000583, standard compositions A357625-A357626.
The version for original alternating sum is A345958.
Positions of ones in A357633, non-reverse A357629.
The skew version for k = 0 is A357636, non-reverse A357632.
These partitions are counted by A035444, skew A035544.
The non-reverse version is A357851, k = 0 version A357631.
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-length A357642.
Cf. A000290, A003963, A053251, A055932, A357621-A357624, A357630, A357634, A357637, A357639, A357640.

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

A357645 Triangle read by rows where T(n,k) is the number of integer compositions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 2, 2, 4, 0, 0, 3, 5, 3, 5, 0, 0, 4, 8, 10, 4, 6, 0, 0, 5, 11, 18, 18, 5, 7, 0, 0, 6, 14, 28, 36, 30, 6, 8, 0, 0, 7, 17, 41, 63, 65, 47, 7, 9, 0, 0, 8, 20, 58, 104, 126, 108, 70, 8, 10, 0, 0, 9, 23, 80, 164, 230, 230, 168, 100, 9, 11

Offset: 0

Gus Wiseman, Oct 12 2022

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 - ...

			Triangle begins:
   1
   0   1
   0   0   2
   0   0   1   3
   0   0   2   2   4
   0   0   3   5   3   5
   0   0   4   8  10   4   6
   0   0   5  11  18  18   5   7
   0   0   6  14  28  36  30   6   8
   0   0   7  17  41  63  65  47   7   9
   0   0   8  20  58 104 126 108  70   8  10
Row n = 6 counts the following compositions:
  (114)   (123)    (132)     (141)  (6)
  (1113)  (213)    (222)     (231)  (15)
  (1122)  (1212)   (312)     (321)  (24)
  (1131)  (1221)   (1311)    (411)  (33)
          (2112)   (2211)           (42)
          (2121)   (3111)           (51)
          (11121)  (11112)
          (11211)  (12111)
                   (21111)
                   (111111)

Row sums are A011782.
For original alternating sum we have A097805, unordered A344651.
Column k = n-4 appears to be A177787.
The case of partitions is A357637, skew A357638.
The central column k=0 is A357641 (aerated).
The skew-alternating version is A357646.
The reverse version for partitions is A357704, skew A357705.
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. A029862, A035363, A035544, A357136, A357631, A357639.

halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],halfats[#]==k&]],{n,0,10},{k,-n,n,2}]

A357704 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 3, 0, 0, 2, 2, 0, 3, 0, 0, 3, 1, 3, 0, 4, 0, 0, 3, 2, 4, 2, 0, 4, 0, 0, 4, 2, 6, 2, 3, 0, 5, 0, 0, 4, 3, 5, 7, 3, 3, 0, 5, 0, 0, 5, 3, 8, 4, 10, 2, 4, 0, 6, 0, 0, 5, 4, 8, 6, 11, 9, 3, 4, 0, 6, 0, 0, 6, 4, 11, 5, 15, 8, 13, 3, 5, 0, 7

Offset: 0

Gus Wiseman, Oct 10 2022

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 - ...

			Triangle begins:
  1
  0  1
  0  0  2
  0  0  1  2
  0  0  2  0  3
  0  0  2  2  0  3
  0  0  3  1  3  0  4
  0  0  3  2  4  2  0  4
  0  0  4  2  6  2  3  0  5
  0  0  4  3  5  7  3  3  0  5
  0  0  5  3  8  4 10  2  4  0  6
  0  0  5  4  8  6 11  9  3  4  0  6
  0  0  6  4 11  5 15  8 13  3  5  0  7
  0  0  6  5 11  8 13 19 10 13  4  5  0  7
  0  0  7  5 14  8 19 13 25  9 17  4  6  0  8
  0  0  7  6 14 11 19 17 29 23 13 18  5  6  0  8
Row n = 7 counts the following reversed partitions:
  .  .  (115)   (124)   (133)      (11113)   .  (7)
        (1114)  (1222)  (223)      (111112)     (16)
        (1123)          (11122)                 (25)
                        (1111111)               (34)

Row sums are A000041.
Last entry of row n is A008619(n).
The central column in the non-reverse case is A035363, skew A035544.
For original reverse-alternating sum we have A344612.
For original alternating sum we have A344651, ordered A097805.
The non-reverse version is A357637, skew A357638.
The central column is A357639, skew A357640.
The non-reverse ordered version (compositions) is A357645, skew A357646.
The skew-alternating version is A357705.
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. A029862, A053251, A357136, A357189, A357487, A357488, A357631, A357632, A357641.

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

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}]

A357851 Numbers k such that the half-alternating sum of the prime indices of k is 1.

Original entry on oeis.org

2, 8, 18, 32, 45, 50, 72, 98, 105, 128, 162, 180, 200, 231, 242, 275, 288, 338, 392, 420, 429, 450, 455, 512, 578, 648, 663, 720, 722, 800, 833, 882, 924, 935, 968, 969, 1050, 1058, 1100, 1125, 1152, 1235, 1250, 1311, 1352, 1458, 1463, 1568, 1680, 1682, 1716

Offset: 1

Gus Wiseman, Oct 28 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 - ...

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:
     2: {1}
     8: {1,1,1}
    18: {1,2,2}
    32: {1,1,1,1,1}
    45: {2,2,3}
    50: {1,3,3}
    72: {1,1,1,2,2}
    98: {1,4,4}
   105: {2,3,4}
   128: {1,1,1,1,1,1,1}
   162: {1,2,2,2,2}
   180: {1,1,2,2,3}
   200: {1,1,1,3,3}

The version for k = 0 is A357631, standard compositions A357625-A357626.
The version for original alternating sum is A001105.
Positions of ones in A357629, reverse A357633.
The skew version for k = 0 is A357632, reverse A357636.
Partitions with these Heinz numbers are counted by A035444, skew A035544.
The reverse version is A357635, k = 0 version A000583.
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-length A357642.
Cf. A003963, A053251, A055932, A345958, A357621-A357624, A357630, A357634, A357637, A357639, A357640.

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

cp's OEIS Frontend

A357626 Numbers k such that the reversed k-th composition in standard order has half-alternating sum 0.

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A357630 Skew-alternating sum of the prime indices of n.

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A357634 Skew-alternating sum of the partition having Heinz number n.

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A357622 Half-alternating sum of the reversed n-th composition in standard order.

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A357635 Numbers k such that the half-alternating sum of the partition having Heinz number k is 1.

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A357645 Triangle read by rows where T(n,k) is the number of integer compositions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.

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A357704 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.

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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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A357851 Numbers k such that the half-alternating sum of the prime indices of k is 1.

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