A175342 -id:A175342 - cp's OEIS frontend

A325547 Number of compositions of n with strictly increasing differences.

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

A342495 Number of compositions of n with constant (equal) first quotients.

Original entry on oeis.org

1, 1, 2, 4, 5, 6, 8, 10, 10, 11, 12, 12, 16, 16, 18, 20, 19, 18, 22, 22, 24, 28, 24, 24, 30, 27, 30, 30, 34, 30, 38, 36, 36, 36, 36, 40, 43, 40, 42, 46, 48, 42, 52, 46, 48, 52, 48, 48, 56, 55, 54, 54, 58, 54, 60, 58, 64, 64, 60, 60, 72, 64, 68, 74, 69, 72, 72

Offset: 0

Gus Wiseman, Mar 17 2021

nonn

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 (1,2,4,8) has first quotients (2,2,2) so is counted under a(15).
The composition (4,5,6) has first quotients (5/4,6/5) so is not counted under a(15).
The a(1) = 1 through a(7) = 10 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (12)   (13)    (14)     (15)      (16)
             (21)   (22)    (23)     (24)      (25)
             (111)  (31)    (32)     (33)      (34)
                    (1111)  (41)     (42)      (43)
                            (11111)  (51)      (52)
                                     (222)     (61)
                                     (111111)  (124)
                                               (421)
                                               (1111111)

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for differences instead of quotients is A175342.
The unordered version is A342496, ranked by A342522.
The strict unordered version is A342515.
The distinct version is A342529.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A002843, A003242, A008965, A048004, A059966, A074206, A167606, A253249, A318991, A318992, A325557, A342528.

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

a(n > 0) = 2*A342496(n) - A000005(n).

A325558 Number of compositions of n with equal circular differences up to sign.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 8, 16, 13, 16, 18, 32, 20, 30, 30, 57, 34, 52, 46, 96, 74, 86, 84, 174, 119, 170, 192, 306, 244, 332, 372, 628, 560, 694, 812, 1259, 1228, 1566, 1852, 2696, 2806, 3538, 4260, 5894, 6482, 8098, 9890, 13392, 15049, 18706, 23018, 30298, 35198

Offset: 1

Gus Wiseman, May 11 2019

nonn

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

The circular differences of a composition c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2).

			The a(1) = 1 through a(8) = 16 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)
                    (1111)  (41)     (42)      (43)       (44)
                            (11111)  (51)      (52)       (53)
                                     (222)     (61)       (62)
                                     (1212)    (1111111)  (71)
                                     (2121)               (1232)
                                     (111111)             (1313)
                                                          (2123)
                                                          (2222)
                                                          (2321)
                                                          (3131)
                                                          (3212)
                                                          (11111111)

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A008965, A047966, A049988, A098504, A173258, A175342, A325553, A325557, A325588, A325589.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Abs[Differences[Append[#,First[#]]]]&]],{n,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,k,s)={my(R=matrix(n,n,i,j,i==j&&abs(i-k)==s), t=0); while(R, R=step(R,n,s); t+=R[n,k]); t}
a(n) = {numdiv(max(1,n)) + sum(s=1, n-1, sum(k=1, n, w(n,k,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

A325849 Number of strict compositions of n with no three consecutive parts in arithmetic progression.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 13, 19, 23, 51, 57, 91, 117, 179, 283, 381, 531, 737, 1017, 1335, 2259, 2745, 3983, 5289, 7367, 9413, 13155, 19461, 25129, 33997, 45633, 61225, 80481, 107091, 137475, 205243, 253997, 345527, 447003, 604919, 768331, 1026167, 1299227

Offset: 0

Gus Wiseman, May 31 2019

nonn

A composition of n is a finite sequence of positive integers with sum n. a(n) is the number of strict compositions of n with no two of their adjacent first-differences equal, or with no 0's in their second-differences.

			The a(1) = 1 through a(8) = 19 compositions:
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)
            (12)  (13)  (14)  (15)   (16)   (17)
            (21)  (31)  (23)  (24)   (25)   (26)
                        (32)  (42)   (34)   (35)
                        (41)  (51)   (43)   (53)
                              (132)  (52)   (62)
                              (213)  (61)   (71)
                              (231)  (124)  (125)
                              (312)  (142)  (134)
                                     (214)  (143)
                                     (241)  (152)
                                     (412)  (215)
                                     (421)  (251)
                                            (314)
                                            (341)
                                            (413)
                                            (431)
                                            (512)
                                            (521)

Wikipedia, Arithmetic progression

The non-strict case is A238423.

Cf. A007862, A049988, A175342, A279945, A295370, A325545, A325874, A325875.

Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],!MemberQ[Differences[#,2],0]&]],{n,0,30}]

A342515 Number of strict partitions of n with constant (equal) first-quotients.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 4, 5, 5, 6, 6, 8, 8, 9, 8, 9, 9, 11, 10, 13, 11, 12, 12, 13, 14, 14, 15, 15, 16, 18, 16, 17, 17, 19, 18, 20, 20, 22, 21, 21, 23, 23, 22, 24, 23, 24, 24, 27, 25, 26, 27, 27, 27, 28, 29, 31, 29, 30, 31, 32, 33, 35, 32, 35, 33, 35, 34, 35

Offset: 0

Gus Wiseman, Mar 19 2021

nonn

Also the number of reversed strict partitions of n with constant (equal) first-quotients.

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

			The a(1) = 1 through a(15) = 9 partitions (A..F = 10..15):
  1   2   3    4    5    6    7     8    9    A    B    C    D     E     F
          21   31   32   42   43    53   54   64   65   75   76    86    87
                    41   51   52    62   63   73   74   84   85    95    96
                              61    71   72   82   83   93   94    A4    A5
                              421        81   91   92   A2   A3    B3    B4
                                                   A1   B1   B2    C2    C3
                                                             C1    D1    D2
                                                             931   842   E1
                                                                         8421

Wikipedia, Arithmetic progression
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for differences instead of quotients is A049980.
The non-strict ordered version is A342495.
The non-strict version is A342496.
The distinct instead of equal version is A342520.
A000005 counts constant partitions.
A000041 counts partitions (strict: A000009).
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A154402 counts partitions with adjacent parts x = 2y.
A167865 counts strict chains of divisors > 1 summing to n.
A175342 counts compositions with equal differences.
Cf. A003242, A005117, A049988, A057567, A067824, A253249, A307824, A318991, A318992, A325328, A342086.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&SameQ@@Divide@@@Partition[#,2,1]&]],{n,0,30}]

A325852 Number of (strict) integer partitions of n whose differences of all degrees are nonzero.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 11, 15, 19, 19, 26, 31, 31, 41, 49, 53, 62, 75, 81, 97, 112, 124, 145, 171, 175, 215, 244, 274, 307, 344, 388, 446, 497, 561, 599, 700, 779, 881, 981, 1054, 1184, 1340, 1500, 1669, 1767, 2031, 2237, 2486, 2765, 2946, 3300

Offset: 0

Gus Wiseman, May 31 2019

nonn

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. The differences of all degrees of a sequence are the union of its zeroth through m-th differences, where m is the length of the sequence.

			The a(1) = 1 through a(11) = 11 partitions (A = 10, B = 11):
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)   (65)
                        (41)  (51)  (52)   (62)   (63)   (73)   (74)
                                    (61)   (71)   (72)   (82)   (83)
                                    (421)  (431)  (81)   (91)   (92)
                                           (521)  (621)  (532)  (A1)
                                                         (541)  (542)
                                                         (631)  (632)
                                                         (721)  (641)
                                                                (731)
                                                                (821)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..250

The case for only degrees > 1 is A325874.

Cf. A049988, A175342, A238423, A279945, A295370, A325328, A325468, A325545, A325850, A325851, A325875.

Table[Length[Select[IntegerPartitions[n],!MemberQ[Union@@Table[Differences[#,i],{i,Length[#]}],0]&]],{n,0,30}]

A342496 Number of integer partitions of n with constant (equal) first quotients.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 6, 7, 7, 8, 7, 11, 9, 11, 12, 12, 10, 14, 12, 15, 16, 14, 13, 19, 15, 17, 17, 20, 16, 23, 19, 21, 20, 20, 22, 26, 21, 23, 25, 28, 22, 30, 24, 27, 29, 26, 25, 33, 29, 30, 29, 32, 28, 34, 31, 36, 34, 32, 31, 42

Offset: 0

Gus Wiseman, Mar 17 2021

nonn

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 partition (12,6,3) has first quotients (1/2,1/2) so is counted under a(21).
The a(1) = 1 through a(9) = 7 partitions:
  1   2    3     4      5       6        7         8          9
      11   21    22     32      33       43        44         54
           111   31     41      42       52        53         63
                 1111   11111   51       61        62         72
                                222      421       71         81
                                111111   1111111   2222       333
                                                   11111111   111111111

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for differences instead of quotients is A049988.
The ordered version is A342495.
The distinct version is A342514.
The strict case is A342515.
The Heinz numbers of these partitions are A342522.
A000005 counts constant partitions.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A000837, A002843, A003242, A074206, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Table[Length[Select[IntegerPartitions[n],SameQ@@Divide@@@Partition[#,2,1]&]],{n,0,30}]

a(n > 0) = (A342495(n) + A000005(n))/2.

A325850 Number of permutations of {1..n} whose differences of all degrees are nonzero.

Original entry on oeis.org

1, 1, 2, 4, 18, 72, 446, 2804, 21560, 184364, 1788514

Offset: 0

Gus Wiseman, May 31 2019

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. The differences of all degrees of a sequence are the union of its zeroth through m-th differences, where m is the length of the sequence.

			The a(1) = 1 through a(4) = 18 permutations:
  (1)  (12)  (132)  (1243)
       (21)  (213)  (1324)
             (231)  (1342)
             (312)  (1423)
                    (2134)
                    (2143)
                    (2314)
                    (2413)
                    (2431)
                    (3124)
                    (3142)
                    (3241)
                    (3412)
                    (3421)
                    (4132)
                    (4213)
                    (4231)
                    (4312)

Dominated by A295370, the case for only differences of degree 2.

Cf. A049988, A175342, A238423, A279945, A325545, A325851, A325852, A325874, A325875.

Table[Length[Select[Permutations[Range[n]],!MemberQ[Union@@Table[Differences[#,i],{i,Length[#]}],0]&]],{n,0,5}]

A325851 Number of (strict) compositions of n whose differences of all degrees are nonzero.

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 12, 21, 35, 58, 102, 171, 284, 485, 819, 1355, 2301, 3884, 6528, 10983, 18380, 30824, 51851

Offset: 0

Gus Wiseman, May 31 2019

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. The differences of all degrees of a sequence are the union of its zeroth through m-th differences, where m is the length of the sequence.

			The a(1) = 1 through a(7) = 21 compositions:
  (1)  (2)  (3)   (4)    (5)    (6)     (7)
            (12)  (13)   (14)   (15)    (16)
            (21)  (31)   (23)   (24)    (25)
                  (121)  (32)   (42)    (34)
                         (41)   (51)    (43)
                         (131)  (132)   (52)
                         (212)  (141)   (61)
                                (213)   (124)
                                (231)   (142)
                                (312)   (151)
                                (1212)  (214)
                                (2121)  (232)
                                        (241)
                                        (313)
                                        (412)
                                        (421)
                                        (1213)
                                        (1312)
                                        (2131)
                                        (3121)
                                        (12121)

The case for only degrees > 1 is A325875.

Cf. A049988, A175342, A238423, A295370, A325328, A325545, A325850, A325852, A325874.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[Union@@Table[Differences[#,i],{i,Length[#]}],0]&]],{n,0,10}]

cp's OEIS Frontend

A325547 Number of compositions of n with strictly increasing differences.

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A325548 Number of compositions of n with strictly decreasing differences.

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A342495 Number of compositions of n with constant (equal) first quotients.

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A325558 Number of compositions of n with equal circular differences up to sign.

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A325849 Number of strict compositions of n with no three consecutive parts in arithmetic progression.

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A342515 Number of strict partitions of n with constant (equal) first-quotients.

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A325852 Number of (strict) integer partitions of n whose differences of all degrees are nonzero.

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A342496 Number of integer partitions of n with constant (equal) first quotients.

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