A325546 -id:A325546 - cp's OEIS frontend

A325545 Number of compositions of n with distinct differences.

1, 1, 2, 3, 7, 13, 17, 34, 59, 105, 166, 279, 442, 730, 1157, 1927, 3045, 4741, 7527, 11667, 18048, 27928, 43334, 65861, 101385, 153404, 232287, 347643, 523721, 780083, 1165331, 1725966, 2561625, 3773838, 5561577, 8151209, 11920717, 17364461, 25269939, 36635775

Offset: 0

Gus Wiseman, May 10 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(6) = 17 compositions:
  (1)  (2)   (3)   (4)    (5)     (6)
       (11)  (12)  (13)   (14)    (15)
             (21)  (22)   (23)    (24)
                   (31)   (32)    (33)
                   (112)  (41)    (42)
                   (121)  (113)   (51)
                   (211)  (122)   (114)
                          (131)   (132)
                          (212)   (141)
                          (221)   (213)
                          (311)   (231)
                          (1121)  (312)
                          (1211)  (411)
                                  (1131)
                                  (1221)
                                  (1311)
                                  (2112)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..70
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A000740, A008965, A059966, A070211, A175342, A242882, A325325, A325352, A325546, A325547, A325548, A325551, A325554.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Differences[#]&]],{n,0,15}]

More terms from Alois P. Heinz, May 11 2019

A175342 Number of arithmetic progressions (where the difference between adjacent terms is either positive, 0, or negative) of positive integers that sum to n.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 8, 10, 15, 14, 12, 22, 14, 18, 28, 21, 18, 34, 20, 28, 38, 28, 24, 46, 31, 32, 48, 38, 30, 62, 32, 40, 58, 42, 46, 73, 38, 46, 68, 58, 42, 84, 44, 56, 90, 56, 48, 94, 55, 70, 90, 66, 54, 106, 70, 74, 100, 70, 60, 130, 62, 74, 118, 81, 82, 130, 68, 84, 120

Offset: 1

Leroy Quet, Apr 17 2010

nonn

			From _Gus Wiseman_, May 15 2019: (Start)
The a(1) = 1 through a(8) = 10 compositions with equal differences:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (21)   (22)    (23)     (24)      (25)       (26)
             (111)  (31)    (32)     (33)      (34)       (35)
                    (1111)  (41)     (42)      (43)       (44)
                            (11111)  (51)      (52)       (53)
                                     (123)     (61)       (62)
                                     (222)     (1111111)  (71)
                                     (321)                (2222)
                                     (111111)             (11111111)
(End)

Lars Blomberg, Table of n, a(n) for n = 1..10000
Lars Blomberg, C# program for calculating b-file.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000005, A000079, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A070211, A127938, A175327, A325328, A325407, A325545, A325546, A325547, A325548, A325557, A325558.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Differences[#]&]],{n,0,15}] (* returns a(0) = 1, Gus Wiseman, May 15 2019*)

a(n) = 2*A049988(n) - A000005(n).

G.f.: x/(1-x) + Sum_{k>=2} x^k * (1 + x^(k(k-1)/2)) / (1 - x^(k(k-1)/2)) / (1 -x^k).

Edited and extended by Max Alekseyev, May 03 2010

A342528 Number of compositions with alternating parts weakly decreasing (or weakly increasing).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 20, 32, 51, 79, 121, 182, 272, 399, 582, 839, 1200, 1700, 2394, 3342, 4640, 6397, 8771, 11955, 16217, 21878, 29386, 39285, 52301, 69334, 91570, 120465, 157929, 206313, 268644, 348674, 451185, 582074, 748830, 960676, 1229208, 1568716, 1997064

Offset: 0

Gus Wiseman, Mar 24 2021

nonn

These are finite sequences q of positive integers summing to n such that q(i) >= q(i+2) for all possible i.
The strict case (alternating parts are strictly decreasing) is A000041. Is there a bijective proof?
Yes. Construct a Ferrers diagram by placing odd parts horizontally and even parts vertically in a fishbone pattern. The resulting Ferrers diagram will be for an ordinary partition and the process is reversible. It does not appear that this method can be applied to give a formula for this sequence. - Andrew Howroyd, Mar 25 2021

			The a(1) = 1 through a(6) = 20 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (121)   (41)     (42)
                    (211)   (131)    (51)
                    (1111)  (212)    (141)
                            (221)    (222)
                            (311)    (231)
                            (1211)   (312)
                            (2111)   (321)
                            (11111)  (411)
                                     (1212)
                                     (1311)
                                     (2121)
                                     (2211)
                                     (3111)
                                     (12111)
                                     (21111)
                                     (111111)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 501 terms from Andrew Howroyd)
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The even-length case is A114921.
The version with alternating parts unequal is A224958 (unordered: A000726).
The version with alternating parts equal is A342527.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A000203 adds up divisors.
A002843 counts compositions with all adjacent parts x <= 2y.
A003242 counts anti-run compositions.
A069916/A342492 = decreasing/increasing first quotients.
A070211/A325546 = weakly decreasing/increasing differences.
A175342/A325545 = constant/distinct differences.
A342495 = constant first quotients (unordered: A342496, strict: A342515, ranking: A342522).
Cf. A001522, A008965, A048004, A059966, A062968, A064410, A064428, A065608, A167606, A325557, A342519.

b:= proc(n, i, j) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, j)+b(n-i, min(n-i, j), min(n-i, i))))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..42);  # Alois P. Heinz, Jan 16 2025

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],GreaterEqual@@Plus@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]

seq(n)={my(p=1/prod(k=1, n, 1-y*x^k + O(x*x^n))); Vec(1+sum(k=1, n, polcoef(p,k,y)*(polcoef(p,k-1,y) + polcoef(p,k,y))))} \\ Andrew Howroyd, Mar 24 2021

G.f.: Sum_{k>=0} ([y^k] P(x,y))*([y^k] (1 + y)*P(x,y)), where P(x,y) = Product_{k>=1} 1/(1 - y*x^k). - Andrew Howroyd, Jan 16 2025

Terms a(21) and beyond from Andrew Howroyd, Mar 24 2021

A325557 Number of compositions of n with equal differences up to sign.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 13, 12, 20, 24, 25, 29, 49, 40, 50, 64, 86, 80, 105, 102, 164, 175, 186, 208, 325, 316, 382, 476, 624, 660, 814, 961, 1331, 1500, 1739, 2140, 2877, 3274, 3939, 4901, 6345, 7448, 9054, 11157, 14315, 17181, 20769, 25843, 32947, 39639, 48257, 60075

Offset: 0

Gus Wiseman, May 11 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(8) = 20 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (21)   (22)    (23)     (24)      (25)       (26)
             (111)  (31)    (32)     (33)      (34)       (35)
                    (121)   (41)     (42)      (43)       (44)
                    (1111)  (131)    (51)      (52)       (53)
                            (212)    (123)     (61)       (62)
                            (11111)  (141)     (151)      (71)
                                     (222)     (232)      (161)
                                     (321)     (313)      (242)
                                     (1212)    (12121)    (323)
                                     (2121)    (1111111)  (1232)
                                     (111111)             (1313)
                                                          (2123)
                                                          (2222)
                                                          (2321)
                                                          (3131)
                                                          (3212)
                                                          (21212)
                                                          (11111111)

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A047966, A049988, A070211, A098504, A173258, A175342, A325545, A325546, A325547, A325548, A325552, A325558.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Abs[Differences[#]]&]],{n,0,15}]

step(R,n,s)={matrix(n, n, i, j, if(i>j, if(j>s, R[i-j, j-s]) + if(j+s<=n, R[i-j, j+s])) )}
w(n,s)={my(R=matid(n), t=0); while(R, R=step(R,n,s); t+=vecsum(R[n,])); t}
a(n) = {numdiv(max(1,n)) + sum(s=1, n-1, w(n,s))} \\ Andrew Howroyd, Aug 22 2019

a(26)-a(42) from Lars Blomberg, May 30 2019

Terms a(43) and beyond from Andrew Howroyd, Aug 22 2019

A070211 Number of compositions (ordered partitions) of n that are concave-down sequences.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 14, 18, 24, 34, 42, 52, 68, 82, 101, 126, 147, 175, 213, 246, 289, 344, 392, 453, 530, 598, 687, 791, 885, 1007, 1151, 1276, 1438, 1629, 1806, 2018, 2262, 2490, 2775, 3091, 3387, 3754, 4165, 4542, 5011, 5527, 6012, 6600, 7245, 7864, 8614

Offset: 0

Pontus von Brömssen, May 07 2002

Here, a finite sequence is concave if each term (other than the first or last) is at least the average of the two adjacent terms. - Eric M. Schmidt, Sep 29 2013

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1). Then a(n) is the number of compositions of n with weakly decreasing differences. - Gus Wiseman, May 15 2019

			Out of the 8 ordered partitions of 4, only 2+1+1 and 1+1+2 are not concave, so a(4)=6.
From _Gus Wiseman_, May 15 2019: (Start)
The a(1) = 1 through a(6) = 14 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (121)   (41)     (42)
                    (1111)  (122)    (51)
                            (131)    (123)
                            (221)    (132)
                            (11111)  (141)
                                     (222)
                                     (231)
                                     (321)
                                     (1221)
                                     (111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..500
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A001523 (weakly unimodal compositions), A069916, A175342, A320466, A325361 (concave-down partitions), A325545, A325546 (concave-up compositions), A325547, A325548, A325557.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],GreaterEqual@@Differences[#]&]],{n,0,15}] (* Gus Wiseman, May 15 2019 *)

def A070211(n) : return sum(all(2*p[i] >= p[i-1] + p[i+1] for i in range(1, len(p)-1)) for p in Compositions(n)) # Eric M. Schmidt, Sep 29 2013

Name edited by Gus Wiseman, May 15 2019

A325547 Number of compositions of n with strictly increasing differences.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 11, 18, 24, 30, 45, 57, 71, 96, 120, 148, 192, 235, 286, 354, 431, 518, 628, 752, 893, 1063, 1262, 1482, 1744, 2046, 2386, 2775, 3231, 3733, 4305, 4977, 5715, 6536, 7507, 8559, 9735, 11112, 12608, 14252, 16177, 18265, 20553, 23204, 26090, 29223

Offset: 0

Gus Wiseman, May 10 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(6) = 11 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)
       (11)  (12)  (13)   (14)   (15)
             (21)  (22)   (23)   (24)
                   (31)   (32)   (33)
                   (112)  (41)   (42)
                   (211)  (113)  (51)
                          (212)  (114)
                          (311)  (213)
                                 (312)
                                 (411)
                                 (2112)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A000740, A008965, A034297, A070211, A175342, A179269, A179254, A240027, A325545, A325546, A325548, A325552, A325557.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Less@@Differences[#]&]],{n,0,15}]

\\ Row sums of R(n) give A179269 (breakdown by width)
R(n)={my(L=List(), v=vectorv(n, i, 1), w=1, t=1); while(v, listput(L,v); w++; t+=w; v=vectorv(n, i, sum(k=1, (i-1)\t, v[i-k*t]))); Mat(L)}
seq(n)={my(M=R(n)); Vec(1 + sum(i=1, n, my(p=sum(w=1, min(#M,n\i), x^(w*i)*sum(j=1, n-i*w, x^j*M[j,w])));  x^i*(1 + x^i)*(1 + p + O(x*x^(n-i)))^2))} \\ Andrew Howroyd, Aug 27 2019

a(26)-a(42) from Lars Blomberg, May 30 2019

Terms a(43) and beyond from Andrew Howroyd, Aug 27 2019

A325548 Number of compositions of n with strictly decreasing differences.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 10, 13, 19, 23, 29, 38, 46, 55, 69, 80, 96, 115, 132, 154, 183, 207, 238, 276, 314, 356, 405, 455, 513, 579, 647, 724, 809, 897, 998, 1107, 1225, 1350, 1486, 1639, 1805, 1973, 2166, 2374, 2586, 2824, 3084, 3346, 3646, 3964, 4286, 4655, 5047

Offset: 0

Gus Wiseman, May 10 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(8) = 19 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)    (8)
       (11)  (12)  (13)   (14)   (15)    (16)   (17)
             (21)  (22)   (23)   (24)    (25)   (26)
                   (31)   (32)   (33)    (34)   (35)
                   (121)  (41)   (42)    (43)   (44)
                          (122)  (51)    (52)   (53)
                          (131)  (132)   (61)   (62)
                          (221)  (141)   (133)  (71)
                                 (231)   (142)  (134)
                                 (1221)  (151)  (143)
                                         (232)  (152)
                                         (241)  (161)
                                         (331)  (233)
                                                (242)
                                                (251)
                                                (332)
                                                (341)
                                                (431)
                                                (1331)

Alois P. Heinz, Table of n, a(n) for n = 0..400
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A011782, A000740, A008965, A070211, A175342, A179254, A320470, A325457, A325545, A325546, A325547, A325552, A325557.

b:= proc(n, l, d) option remember; `if`(n=0, 1, add(`if`(l=0 or
       j-l b(n, 0$2):
seq(a(n), n=0..52);  # Alois P. Heinz, Jan 27 2024

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Greater@@Differences[#]&]],{n,0,15}]

a(26)-a(44) from Lars Blomberg, May 30 2019

a(45)-a(52) from Alois P. Heinz, Jan 27 2024

A068322 Number of arithmetic progressions of positive odd integers, strictly increasing with sum n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 3, 3, 5, 1, 4, 1, 5, 4, 5, 1, 7, 2, 6, 5, 8, 1, 7, 1, 9, 6, 8, 2, 11, 1, 9, 7, 12, 1, 10, 1, 12, 10, 11, 1, 15, 2, 12, 9, 15, 1, 13, 3, 16, 10, 14, 1, 18, 1, 15, 12, 20, 4, 17, 1, 19, 12, 17, 1, 22, 1, 18, 16, 22, 2, 20, 1, 24, 15, 20, 1, 25, 5, 21, 15, 26

Offset: 1

Naohiro Nomoto, Feb 27 2002

			From _Petros Hadjicostas_, Sep 29 2019: (Start)
a(12) = 3 because we have the following arithmetic progressions of odd numbers, strictly increasing with sum n=12: 1+11, 3+9, and 5+7.
a(13) = 1 because we have only the following arithmetic progressions of odd numbers, strictly increasing with sum n=13: 13.
a(14) = 3 because we have the following arithmetic progressions of odd numbers, strictly increasing with sum n=14: 1+13, 3+11, and 5+9.
a(15) = 3 because we have the following arithmetic progressions of odd numbers, strictly increasing with sum n=15: 15, 3+5+7, and 1+5+9.
(End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A068323, A068324, A070211, A127938, A175327, A325328, A325407, A325545, A325546, A325547, A325548.

From Petros Hadjicostas, Oct 01 2019: (Start)
a(n) = A068324(n) - A001227(n) + (1/2) * (1 - (-1)^n).
G.f.: x/(1 - x^2) + Sum_{m >= 2} x^(m^2)/((1 - x^(2*m)) * (1 - x^(m*(m-1)))).
(End)

A342492 Number of compositions of n with weakly increasing first quotients.

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 17, 26, 37, 52, 73, 95, 125, 163, 208, 261, 330, 407, 498, 607, 734, 881, 1056, 1250, 1480, 1738, 2029, 2359, 2742, 3160, 3635, 4169, 4760, 5414, 6151, 6957, 7861, 8858, 9952, 11148, 12483, 13934, 15526, 17267, 19173, 21252, 23535, 25991

Offset: 0

Gus Wiseman, Mar 16 2021

nonn

Also called log-concave-up compositions.

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The composition (4,2,1,2,3) has first quotients (1/2,1/2,2,3/2) so is not counted under a(12), even though the first differences (-2,-1,1,1) are weakly increasing.
The a(1) = 1 through a(6) = 17 compositions:
  (1)  (2)    (3)      (4)        (5)          (6)
       (1,1)  (1,2)    (1,3)      (1,4)        (1,5)
              (2,1)    (2,2)      (2,3)        (2,4)
              (1,1,1)  (3,1)      (3,2)        (3,3)
                       (1,1,2)    (4,1)        (4,2)
                       (2,1,1)    (1,1,3)      (5,1)
                       (1,1,1,1)  (2,1,2)      (1,1,4)
                                  (3,1,1)      (2,1,3)
                                  (1,1,1,2)    (2,2,2)
                                  (2,1,1,1)    (3,1,2)
                                  (1,1,1,1,1)  (4,1,1)
                                               (1,1,1,3)
                                               (2,1,1,2)
                                               (3,1,1,1)
                                               (1,1,1,1,2)
                                               (2,1,1,1,1)
                                               (1,1,1,1,1,1)

Alois P. Heinz, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The weakly decreasing version is A069916.
The version for differences instead of quotients is A325546.
The strictly increasing version is A342493.
The unordered version is A342497, ranked by A342523.
The strict unordered version is A342516.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations.
A002843 counts compositions with all adjacent parts x <= 2y.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A003242, A008965, A048004, A059966, A067824, A167606, A253249, A274199, A318991, A318992, A342527, A342528.

b:= proc(n, q, l) option remember; `if`(n=0, 1, add(
     `if`(q=0 or q>=l/j, b(n-j, l/j, j), 0), j=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 25 2021

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],LessEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]
(* Second program: *)
b[n_, q_, l_] := b[n, q, l] = If[n == 0, 1, Sum[
     If[q == 0 || q >= l/j, b[n - j, l/j, j], 0], {j, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

a(21)-a(47) from Alois P. Heinz, Mar 25 2021

A342532 Number of even-length compositions of n with alternating parts distinct.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 9, 14, 28, 44, 83, 136, 250, 424, 757, 1310, 2313, 4018, 7081, 12314, 21650, 37786, 66264, 115802, 202950, 354858, 621525, 1087252, 1903668, 3330882, 5831192, 10204250, 17862232, 31260222, 54716913, 95762576, 167614445, 293356422, 513456686

Offset: 0

Gus Wiseman, Mar 28 2021

nonn

These are finite even-length sequences q of positive integers summing to n such that q(i) != q(i+2) for all possible i.

			The a(2) = 1 through a(7) = 14 compositions:
  (1,1)  (1,2)  (1,3)  (1,4)  (1,5)      (1,6)
         (2,1)  (2,2)  (2,3)  (2,4)      (2,5)
                (3,1)  (3,2)  (3,3)      (3,4)
                       (4,1)  (4,2)      (4,3)
                              (5,1)      (5,2)
                              (1,1,2,2)  (6,1)
                              (1,2,2,1)  (1,1,2,3)
                              (2,1,1,2)  (1,1,3,2)
                              (2,2,1,1)  (1,2,3,1)
                                         (1,3,2,1)
                                         (2,1,1,3)
                                         (2,3,1,1)
                                         (3,1,1,2)
                                         (3,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..500

The strictly decreasing version appears to be A064428 (odd-length: A001522).
The equal version is A065608 (A342527 with odds).
The weakly decreasing version is A114921 (A342528 with odds).
Including odds gives A224958.
A000726 counts partitions with alternating parts unequal.
A325545 counts compositions with distinct first differences.
A342529 counts compositions with distinct first quotients.
Cf. A000009, A000041, A003242, A008965, A032020, A059966, A062968, A064410, A070211, A106351, A175342, A325546.

qdq[q_]:=And@@Table[q[[i]]!=q[[i+2]],{i,Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],EvenQ[Length[#]]&],qdq]],{n,0,15}]

\\ here gf gives A106351 as g.f.
gf(n, y)={1/(1 - sum(k=1, n, (-1)^(k+1)*x^k*y^k/(1-x^k) + O(x*x^n)))}
seq(n)={my(p=gf(n,y)); Vec(sum(k=0, n\2, polcoef(p,k,y)^2))} \\ Andrew Howroyd, Apr 16 2021

G.f.: 1 + Sum_{k>=1} B_k(x)^2 where B_k(x) is the g.f. of column k of A106351. - Andrew Howroyd, Apr 16 2021

Terms a(24) and beyond from Andrew Howroyd, Apr 16 2021

cp's OEIS Frontend

A325545 Number of compositions of n with distinct differences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A175342 Number of arithmetic progressions (where the difference between adjacent terms is either positive, 0, or negative) of positive integers that sum to n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A342528 Number of compositions with alternating parts weakly decreasing (or weakly increasing).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A325557 Number of compositions of n with equal differences up to sign.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A070211 Number of compositions (ordered partitions) of n that are concave-down sequences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Extensions

A325547 Number of compositions of n with strictly increasing differences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A325548 Number of compositions of n with strictly decreasing differences.

Views

Author

Keywords

Comments

Examples

Links