A344616 -id:A344616 - cp's OEIS frontend

A344608 Number of integer partitions of n with reverse-alternating sum < 0.

0, 0, 0, 1, 1, 3, 3, 7, 7, 14, 15, 27, 29, 49, 54, 86, 96, 146, 165, 242, 275, 392, 449, 623, 716, 973, 1123, 1498, 1732, 2274, 2635, 3411, 3955, 5059, 5871, 7427, 8620, 10801, 12536, 15572, 18065, 22267, 25821, 31602, 36617, 44533, 51560, 62338, 72105, 86716

Offset: 0

Gus Wiseman, May 30 2021

nonn

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
Also the number of reversed of integer partitions of n with alternating sum < 0.
No integer partitions have alternating sum < 0, so the non-reversed version is all zeros.
Is this sequence weakly increasing? Note: a(2n + 2) = A236914(n), a(2n) = A344743(n).
A formula for the reverse-alternating sum of a partition is: (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of integer partitions of n of even length whose conjugate parts are not all odd. Partitions of the latter type are counted by A086543. By conjugation, a(n) is also the number of integer partitions of n of even maximum whose parts are not all odd.

			The a(3) = 1 through a(9) = 14 partitions:
  (21)  (31)  (32)    (42)    (43)      (53)      (54)
              (41)    (51)    (52)      (62)      (63)
              (2111)  (3111)  (61)      (71)      (72)
                              (2221)    (3221)    (81)
                              (3211)    (4211)    (3222)
                              (4111)    (5111)    (3321)
                              (211111)  (311111)  (4221)
                                                  (4311)
                                                  (5211)
                                                  (6111)
                                                  (222111)
                                                  (321111)
                                                  (411111)
                                                  (21111111)

The opposite version (rev-alt sum > 0) is A027193, ranked by A026424.
The strict case (for n > 2) is A067659 (odd bisection: A344650).
The Heinz numbers of these partitions are A119899 (complement: A344609).
The bisections are A236914 (odd) and A344743 (even).
The ordered version appears to be A294175 (even bisection: A008549).
The complement is counted by A344607 (even bisection: A344611).
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A027187 counts partitions with alternating sum <= 0, ranked by A028260.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions with rev-alternating sum 2 (negative: A344741).
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.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344618 gives reverse-alternating sums of standard compositions.
Cf. A000070, A000097, A000346, A003242, A006330, A071322, A239829, A239830, A344649, A344651, A344654/A344740, A344739.

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

A347438 Number of unordered factorizations of n with alternating product 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Sep 06 2021

nonn

Also the number of unordered factorizations of n with alternating sum 0.
Also the number of unordered factorizations of n with all even multiplicities.
This is the even-length case of A347437, the odd-length case being A347441.
An unordered factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			The a(n) factorizations for n = 16, 64, 144, 256, 576:
  4*4      8*8          12*12        16*16            24*24
  2*2*2*2  2*2*4*4      2*2*6*6      2*2*8*8          3*3*8*8
           2*2*2*2*2*2  3*3*4*4      4*4*4*4          4*4*6*6
                        2*2*2*2*3*3  2*2*2*2*4*4      2*2*12*12
                                     2*2*2*2*2*2*2*2  2*2*2*2*6*6
                                                      2*2*3*3*4*4
                                                      2*2*2*2*2*2*3*3

Antti Karttunen, Table of n, a(n) for n = 1..65537
PlanetMath, alternating sum

Positions of zeros are A000037.
Positions of nonzero terms are A000290.
The restriction to perfect squares is A001055 (ordered: A273013).
The restriction to powers of 2 is A035363.
The additive version is A119620, ranked by A028982.
Positions of non-1's are A213367 \ {1}.
Positions of 1's are A280076 = {1} \/ A001248.
Sorted first positions are 1, 2, and all terms of A330972 squared.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
Allowing any integer alternating product gives A347437.
Allowing any integer reciprocal alternating product gives A347439.
Allowing any alternating product < 1 gives A347440.
Allowing any alternating product >= 1 gives A347456.
A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344606 counts alternating permutations of prime factors.
A347441 counts odd-length factorizations with integer alternating product.
A347460 counts possible alternating products of factorizations.
Cf. A000041, A005117, A025047, A038548, A062312, A088218, A316523, A332269, A344607, A347442, A347446, A347463.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[facs[n],altprod[#]==1&]],{n,100}]

A347438(n, m=n, k=0, t=1) = if(1==n, (1==t), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A347438(n/d, d, 1-k, t*(d^((-1)^k))))); (s)); \\ Antti Karttunen, Oct 30 2021

a(2^n) = A035363(n).

a(n^2) = A001055(n).

Name and comments clarified (with unordered) by Jacob Sprittulla, Oct 05 2021

A163493 Number of binary strings of length n which have the same number of 00 and 01 substrings.

Original entry on oeis.org

1, 2, 2, 3, 6, 9, 15, 30, 54, 97, 189, 360, 675, 1304, 2522, 4835, 9358, 18193, 35269, 68568, 133737, 260802, 509132, 995801, 1948931, 3816904, 7483636, 14683721, 28827798, 56637969, 111347879, 219019294, 431043814, 848764585, 1672056525, 3295390800, 6497536449

Offset: 0

Christopher Carl Heckman, Jul 29 2009

nonn

A variation of problem 11424 in the American Mathematical Monthly. Terms were brute-force calculated using Maple 10.
Proposed Problem 11610 in the Dec 2011 A.M.M.
From Gus Wiseman, Jul 27 2021: (Start)
Also the antidiagonal sums of the matrices counting integer compositions by length and alternating sum (A345197). So a(n) is the number of integer compositions of n + 1 of length (n - s + 3)/2, where s is the alternating sum of the composition. For example, the a(0) = 1 through a(6) = 7 compositions are:
  (1)  (2)   (3)   (4)    (5)     (6)     (7)
       (11)  (21)  (31)   (41)    (51)    (61)
                   (121)  (122)   (123)   (124)
                          (221)   (222)   (223)
                          (1112)  (321)   (322)
                          (1211)  (1122)  (421)
                                  (1221)  (1132)
                                  (2112)  (1231)
                                  (2211)  (2122)
                                          (2221)
                                          (3112)
                                          (3211)
                                          (11131)
                                          (12121)
                                          (13111)
For a bijection with the main (binary string) interpretation, take the run-lengths of each binary string of length n + 1 that satisfies the condition and starts with 1.
(End)

			1 + 2*x + 2*x^2 + 3*x^3 + 6*x^4 + 9*x^5 + 15*x^6 + 30*x^7 + 54*x^8 + 97*x^9 + ...
From _Gus Wiseman_, Jul 27 2021: (Start)
The a(0) = 1 though a(6) = 15 binary strings:
  ()  (0)  (1,0)  (0,0,1)  (0,0,1,0)  (0,0,1,1,0)  (0,0,0,1,0,1)
      (1)  (1,1)  (1,1,0)  (0,0,1,1)  (0,0,1,1,1)  (0,0,1,0,0,1)
                  (1,1,1)  (0,1,0,0)  (0,1,1,0,0)  (0,0,1,1,1,0)
                           (1,0,0,1)  (1,0,0,1,0)  (0,0,1,1,1,1)
                           (1,1,1,0)  (1,0,0,1,1)  (0,1,0,0,0,1)
                           (1,1,1,1)  (1,0,1,0,0)  (0,1,1,1,0,0)
                                      (1,1,0,0,1)  (1,0,0,1,1,0)
                                      (1,1,1,1,0)  (1,0,0,1,1,1)
                                      (1,1,1,1,1)  (1,0,1,1,0,0)
                                                   (1,1,0,0,1,0)
                                                   (1,1,0,0,1,1)
                                                   (1,1,0,1,0,0)
                                                   (1,1,1,0,0,1)
                                                   (1,1,1,1,1,0)
                                                   (1,1,1,1,1,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..3328 (first 501 terms from R. H. Hardin)
Shalosh B. Ekhad and Doron Zeilberger, Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type, arXiv preprint arXiv:1112.6207 [math.CO], 2011. See subpages for rigorous derivations of g.f., recurrence, asymptotics for this sequence. [From _N. J. A. Sloane_, Apr 07 2012]
R. Stanley, Problem 11610, Amer. Math. Monthly, 118 (2011), 937; 120 (2013), 943-944.

Antidiagonal sums of the matrices A345197.
Row sums of A345907.
Taking diagonal instead of antidiagonal sums gives A345908.
A011782 counts compositions (or binary strings).
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 (reverse: A344616).
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. A000041, A000070, A000096, A000097, A000124, A000346, A007318, A008549, A025047, A131577, A238279.

with(combinat): count := proc(n) local S, matches, A, k, i; S := subsets(\{seq(i, i=1..n)\}): matches := 0: while not S[finished] do A := S[nextvalue](): k := 0: for i from 1 to n-1 do: if not (i in A) and not (i+1 in A) then k := k + 1: fi: if not (i in A) and (i+1 in A) then k := k - 1: fi: od: if (k = 0) then matches := matches + 1: fi: end do; return(matches); end proc:
# second Maple program:
b:= proc(n, l, t) option remember; `if`(n-abs(t)<0, 0, `if`(n=0, 1,
      add(b(n-1, i, t+`if`(l=0, (-1)^i, 0)), i=0..1)))
    end:
a:= n-> b(n, 1, 0):
seq(a(n), n=0..36);  # Alois P. Heinz, Mar 20 2024

a[0] = 1; a[n_] := Sum[Binomial[2*k - 1, k]*Binomial[n - 2*k, k] + Binomial[2*k, k]*Binomial[n - 2*k - 1, k], {k, 0, n/3}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 28 2017, after Joel B. Lewis *)
Table[Length[Select[Tuples[{0,1},n],Count[Partition[#,2,1],{0,0}]==Count[Partition[#,2,1],{0,1}]&]],{n,0,10}] (* Gus Wiseman, Jul 27 2021 *)
a[0]:=1; a[n_]:=(1 + 3*HypergeometricPFQ[{1/2, 1-3*n/8, (1-n)/3, (2-n)/3, -n/3},{1, (1-n)/2, 1-n/2, -3*n/8}, -27])/2; Array[a,37,0] (* Stefano Spezia, Apr 26 2024 *)

from math import comb
def A163493(n): return 2+sum((x:=comb((k:=m<<1)-1,m)*comb(n-k,m))+(x*(n-3*m)<<1)//(n-k) for m in range(1,n//3+1)) if n else 1 # Chai Wah Wu, May 01 2024

G.f.: 1/2/(1-x) + (1+2*x)/2/sqrt((1-x)*(1-2*x)*(1+x+2*x^2)). - Richard Stanley, corrected Apr 29 2011
G.f.: (1 + sqrt( 1 + 4*x / ((1 - x) * (1 - 2*x) * (1 + x + 2*x^2)))) / (2*(1 - x)). - Michael Somos, Jan 30 2012
a(n) = sum( binomial(2*k-1, k)*binomial(n-2*k,k) + binomial(2*k, k)*binomial(n-2*k-1, k), k=0..floor(n/3)). - Joel B. Lewis, May 21 2011
Conjecture: -n*a(n) +(2+n)*a(n-1) +(3n-12)*a(n-2) +(12-n)*a(n-3) +(2n-18)*a(n-4)+(56-12n)*a(n-5) +(8n-40)*a(n-6)=0. - R. J. Mathar, Nov 28 2011
G.f. y = A(x) satisfies x = (1 - x) * (1 - 2*x) * (1 + x + 2*x^2) * y * (y * (1 - x) - 1). - Michael Somos, Jan 30 2012
Sequence a(n) satisfies 0 = a(n) * (n^2-2*n) + a(n-1) * (-3*n^2+8*n-2) + a(n-2) * (3*n^2-10*n+2) + a(n-3) * (-5*n^2+18*n-6) + a(n-4) * (8*n^2-34*n+22) + a(n-5) * (-4*n^2+20*n-16) except if n=1 or n=2. - Michael Somos, Jan 30 2012
a(n) = (1 + 3*hypergeom([1/2, 1-3*n/8, (1-n)/3, (2-n)/3, -n/3],[1, (1-n)/2, 1-n/2, -3*n/8],-27))/2 for n > 0. - Stefano Spezia, Apr 26 2024
a(n) ~ 2^n / sqrt(Pi*n). - Vaclav Kotesovec, Apr 26 2024

A277579 Number of partitions of n for which the number of even parts is equal to the positive alternating sum of the parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 4, 6, 7, 9, 13, 15, 19, 25, 31, 38, 48, 59, 74, 90, 111, 136, 166, 201, 246, 297, 357, 431, 522, 621, 745, 892, 1063, 1263, 1503, 1780, 2109, 2491, 2941, 3463, 4077, 4783, 5616, 6576, 7689, 8981, 10486, 12207, 14209, 16516, 19178, 22231

Offset: 0

Emeric Deutsch and Alois P. Heinz, Oct 20 2016

nonn

In the first Maple program (improvable) AS gives the positive alternating sum of a finite sequence s, EP gives the number of even terms of a finite sequence of positive integers.
For the specified value of n, the second Maple program lists the partitions of n counted by a(n).
Also the number of integer partitions of n with as many even parts as odd parts in the conjugate partition. - Gus Wiseman, Jul 26 2021

			a(9) = 6: [2,1,1,1,1,1,1,1], [3,2,1,1,1,1], [3,3,2,1], [4,2,2,1], [4,3,1,1], [5,4].
a(10) = 7: [1,1,1,1,1,1,1,1,1,1], [3,2,2,1,1,1], [3,3,1,1,1,1], [4,2,1,1,1,1], [4,3,2,1], [5,5], [6,4].
a(11) = 9: [2,1,1,1,1,1,1,1,1,1], [3,2,1,1,1,1,1,1], [3,3,2,1,1,1], [3,3,3,2], [4,2,2,1,1,1], [4,3,1,1,1,1], [5,2,2,2], [5,4,1,1], [6,5].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The sign-sensitive version is A035457 (aerated version of A000009).
Comparing odd parts to odd conjugate parts gives A277103.
Comparing product of parts to product of conjugate parts gives A325039.
Comparing the rev-alt sum to that of the conjugate gives A345196.
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).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A000070, A000097, A000700, A006330, A027187, A027193, A236559, A257991, A325534, A325535, A344607, A344651.

with(combinat): AS := proc (s) options operator, arrow: abs(add((-1)^(i-1)*s[i], i = 1 .. nops(s))) end proc: EP := proc (s) local ct, j: ct := 0: for j to nops(s) do if `mod`(s[j], 2) = 0 then ct := ct+1 else  end if end do: ct end proc: a := proc (n) local P, c, k: P := partition(n): c := 0: for k to nops(P) do if AS(P[k]) = EP(P[k]) then c := c+1 else  end if end do: c end proc: seq(a(n), n = 0 .. 30);
n := 8: with(combinat): AS := proc (s) options operator, arrow: abs(add((-1)^(i-1)*s[i], i = 1 .. nops(s))) end proc: EP := proc (s) local ct, j: ct := 0: for j to nops(s) do if `mod`(s[j], 2) = 0 then ct := ct+1 else  end if end do: ct end proc: P := partition(n): C := {}: for k to nops(P) do if AS(P[k]) = EP(P[k]) then C := `union`(C, {P[k]}) else  end if end do: C;
# alternative Maple program:
b:= proc(n, i, s, t) option remember; `if`(n=0,
      `if`(s=0, 1, 0), `if`(i<1, 0, b(n, i-1, s, t)+
      `if`(i>n, 0, b(n-i, i, s+t*i-irem(i+1, 2), -t))))
    end:
a:= n-> b(n$2, 0, 1):
seq(a(n), n=0..60);

b[n_, i_, s_, t_] := b[n, i, s, t] = If[n == 0, If[s == 0, 1, 0], If[i<1, 0, b[n, i-1, s, t] + If[i>n, 0, b[n-i, i, s + t*i - Mod[i+1, 2], -t]]]]; a[n_] := b[n, n, 0, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 21 2016, translated from Maple *)
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]]; Table[Length[Select[IntegerPartitions[n],Count[#,?EvenQ]==Count[conj[#],?OddQ]&]],{n,0,15}] (* Gus Wiseman, Jul 26 2021 *)

def a(n):
    AS = lambda s: abs(sum((-1)^i*t for i,t in enumerate(s)))
    EP = lambda s: sum((t+1)%2 for t in s)
    return sum(AS(p) == EP(p) for p in Partitions(n))
print([a(n) for n in (0..30)]) # Peter Luschny, Oct 21 2016

A345171 Numbers whose multiset of prime factors has no alternating permutation.

Original entry on oeis.org

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 80, 81, 88, 96, 104, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 270, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336, 343, 344, 351

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

First differs from A335448 in having 270.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
Also Heinz numbers of integer partitions without a wiggly permutation, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   32: {1,1,1,1,1}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   49: {4,4}
   54: {1,2,2,2}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   80: {1,1,1,1,3}
   81: {2,2,2,2}
   88: {1,1,1,5}
   96: {1,1,1,1,1,2}

Removing squares of primes A001248 gives A344653, counted by A344654.
A superset of A335448, which is counted by A325535.
Positions of 0's in A345164.
The partitions with these Heinz numbers are counted by A345165.
The complement is A345172, counted by A345170.
The separable case is A345173, counted by A345166.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions, complement A261983.
A025047 counts alternating or wiggly compositions, directed A025048, A025049.
A325534 counts separable partitions, ranked by A335433.
A344606 counts alternating permutations of prime indices with twins.
A344742 ranks twins and partitions with an alternating permutation.
A345192 counts non-alternating compositions.
Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344604, A344616, A344652, A344740, A345163, A345168, A345193, A345195, A348380, A348609.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[100],Select[Permutations[Flatten[ ConstantArray@@@FactorInteger[#]]],wigQ]=={}&]

A277103 Number of partitions of n for which the number of odd parts is equal to the positive alternating sum of the parts.

Original entry on oeis.org

1, 1, 0, 1, 3, 3, 1, 3, 10, 10, 4, 10, 27, 27, 13, 28, 69, 69, 37, 72, 161, 162, 96, 171, 361, 364, 230, 388, 768, 777, 522, 836, 1581, 1605, 1128, 1739, 3145, 3203, 2345, 3495, 6094, 6225, 4712, 6831, 11511, 11794, 9198, 13010, 21293, 21875, 17496, 24239

Offset: 0

Emeric Deutsch, Oct 18 2016

nonn

It follows by conjugation that the partition statistics "alternating sum" and "number of odd parts" are equidistributed. Consequently, the self-conjugate partitions satisfy the required condition.
In the first Maple program (improvable) AS gives the positive alternating sum of a finite sequence s, OP gives the number of odd terms of a finite sequence of positive integers.
For the specified value of n, the second Maple program lists the partitions of n counted by a(n).
Number of integer partitions of n with the same number of odd parts as their conjugate. - Gus Wiseman, Jun 27 2021

			a(3) = 1 because we have [2,1]. The partitions [3] and [1,1,1] do not qualify.
a(4) = 3 because we have [3,1], [2,2], and [2,1,1]. The partitions [4] and [1,1,1,1] do not qualify.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Comparing even parts to odd conjugate parts gives A277579.
Comparing product of parts to product of conjugate parts gives A325039.
The reverse version is A345196.
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).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A000070, A000097, A000700, A006330, A027187, A027193, A236559, A257991, A325534, A325535, A344607, A344651.

with(combinat): AS := proc (s) options operator, arrow: abs(add((-1)^(i-1)*s[i], i = 1 .. nops(s))) end proc: OP := proc (s) local ct, j: ct := 0: for j to nops(s) do if `mod`(s[j], 2) = 1 then ct := ct+1 else  end if end do: ct end proc: a := proc (n) local P, c, k: P := partition(n): c := 0: for k to nops(P) do if AS(P[k]) = OP(P[k]) then c := c+1 else end if end do: c end proc: seq(a(n), n = 0 .. 50);
n := 8: with(combinat): AS := proc (s) options operator, arrow: abs(add((-1)^(i-1)*s[i], i = 1 .. nops(s))) end proc: OP := proc (s) local ct, j: ct := 0: for j to nops(s) do if `mod`(s[j], 2) = 1 then ct := ct+1 else  end if end do: ct end proc: P := partition(n): C := {}: for k to nops(P) do if AS(P[k]) = OP(P[k]) then C := `union`(C, {P[k]}) else  end if end do: C;
# alternative Maple program:
b:= proc(n, i, s, t) option remember; `if`(n=0,
      `if`(s=0, 1, 0), `if`(i<1, 0, b(n, i-1, s, t)+
      `if`(i>n, 0, b(n-i, i, s+t*i-irem(i, 2), -t))))
    end:
a:= n-> b(n$2, 0, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 19 2016

b[n_, i_, s_, t_] := b[n, i, s, t] = If[n == 0, If[s == 0, 1, 0], If[i<1, 0, b[n, i-1, s, t] + If[i>n, 0, b[n-i, i, s + t*i - Mod[i, 2], -t]]]]; a[n_] := b[n, n, 0, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]]; Table[Length[Select[IntegerPartitions[n],Count[#,?OddQ]==Count[conj[#],?OddQ]&]],{n,0,15}] (* Gus Wiseman, Jun 27 2021 *)

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

Gus Wiseman, May 30 2021

nonn

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.

			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.

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.

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&]

A345910 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum -1.

Original entry on oeis.org

6, 20, 25, 27, 30, 72, 81, 83, 86, 92, 98, 101, 103, 106, 109, 111, 116, 121, 123, 126, 272, 289, 291, 294, 300, 312, 322, 325, 327, 330, 333, 335, 340, 345, 347, 350, 360, 369, 371, 374, 380, 388, 393, 395, 398, 402, 405, 407, 410, 413, 415, 420, 425, 427

Offset: 1

Gus Wiseman, Jul 01 2021

nonn

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

			The sequence of terms together with the corresponding compositions begins:
      6: (1,2)
     20: (2,3)
     25: (1,3,1)
     27: (1,2,1,1)
     30: (1,1,1,2)
     72: (3,4)
     81: (2,4,1)
     83: (2,3,1,1)
     86: (2,2,1,2)
     92: (2,1,1,3)
     98: (1,4,2)
    101: (1,3,2,1)
    103: (1,3,1,1,1)
    106: (1,2,2,2)
    109: (1,2,1,2,1)

These compositions are counted by A001791.
A version using runs of binary digits is A031444.
These are the positions of -1's in A124754.
The opposite (positive 1) version is A345909.
The reverse version is A345912.
The version for alternating sum of prime indices is A345959.
Standard compositions: A000120, A066099, A070939, A124754, A228351, A344618.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A000070 counts partitions of 2n+1 with alternating sum 1, ranked by A001105.
A011782 counts compositions.
A097805 counts compositions by sum and alternating sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
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. A000097, A000346, A008549, A025047, A027187, A031443, A031448, A114121, A119899, A126869, A238279, A344617.

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

A345912 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum -1.

Original entry on oeis.org

5, 18, 23, 25, 29, 68, 75, 78, 81, 85, 90, 95, 98, 103, 105, 109, 114, 119, 121, 125, 264, 275, 278, 284, 289, 293, 298, 303, 308, 315, 318, 322, 327, 329, 333, 338, 343, 345, 349, 356, 363, 366, 369, 373, 378, 383, 388, 395, 398, 401, 405, 410, 415, 418, 423

Offset: 1

Gus Wiseman, Jul 01 2021

nonn

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.

			The sequence of terms together with the corresponding compositions begins:
      5: (2,1)
     18: (3,2)
     23: (2,1,1,1)
     25: (1,3,1)
     29: (1,1,2,1)
     68: (4,3)
     75: (3,2,1,1)
     78: (3,1,1,2)
     81: (2,4,1)
     85: (2,2,2,1)
     90: (2,1,2,2)
     95: (2,1,1,1,1,1)
     98: (1,4,2)
    103: (1,3,1,1,1)
    105: (1,2,3,1)

These compositions are counted by A001791.
These are the positions of -1's in A344618.
The non-reverse version is A345910.
The opposite (positive 1) version is A345911.
The version for Heinz numbers of partitions is A345959.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
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 (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.
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, A001105, A008549, A025047, A031444, A034871, A114121, A126869, A344608, A345958, A345959.

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[#]]==-1&]

A345917 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum > 0.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 9, 11, 14, 16, 17, 18, 19, 21, 22, 23, 26, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 42, 44, 45, 47, 52, 56, 57, 59, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 84, 85, 87, 88, 89, 90, 91, 93, 94, 95, 100, 104, 105, 107

Offset: 1

Gus Wiseman, Jul 08 2021

nonn

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

			The initial terms and the corresponding compositions:
     1: (1)
     2: (2)
     4: (3)
     5: (2,1)
     7: (1,1,1)
     8: (4)
     9: (3,1)
    11: (2,1,1)
    14: (1,1,2)
    16: (5)
    17: (4,1)
    18: (3,2)
    19: (3,1,1)
    21: (2,2,1)
    22: (2,1,2)

The version for Heinz numbers of partitions is A026424.
These compositions are counted by A027306.
These are the positions of terms > 0 in A124754.
The weak (k >= 0) version is A345913.
The reverse-alternating version is A345918.
The opposite (k < 0) version is A345919.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
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 (reverse: A344616).
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, A027193, A032443, A114121, A163493, A344609, A345908.

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

cp's OEIS Frontend

A344608 Number of integer partitions of n with reverse-alternating sum < 0.

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A347438 Number of unordered factorizations of n with alternating product 1.

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PARI

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A163493 Number of binary strings of length n which have the same number of 00 and 01 substrings.

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A277579 Number of partitions of n for which the number of even parts is equal to the positive alternating sum of the parts.

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A345171 Numbers whose multiset of prime factors has no alternating permutation.

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A277103 Number of partitions of n for which the number of odd parts is equal to the positive alternating sum of the parts.

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

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A345910 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum -1.

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