A261982 -id:A261982 - cp's OEIS frontend

A374705 Number of integer compositions of n whose leaders of maximal strictly increasing runs sum to 2.

0, 0, 2, 0, 2, 3, 4, 7, 8, 14, 17, 27, 33, 48, 63, 84, 112, 147, 191, 248, 322, 409, 527, 666, 845, 1062, 1336, 1666, 2079, 2579, 3190, 3936, 4842, 5933, 7259, 8854, 10768, 13074, 15826, 19120, 23048, 27728, 33279, 39879, 47686, 56916, 67818, 80667, 95777, 113552, 134396

Offset: 0

Gus Wiseman, Aug 12 2024

nonn

The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

			The a(0) = 0 through a(9) = 14 compositions:
  .  .  (2)   .  (112)  (23)   (24)    (25)    (26)    (27)
        (11)     (121)  (113)  (114)   (115)   (116)   (117)
                        (131)  (141)   (151)   (161)   (171)
                               (1212)  (1123)  (1124)  (234)
                                       (1213)  (1214)  (1125)
                                       (1231)  (1241)  (1134)
                                       (1312)  (1313)  (1215)
                                               (1412)  (1251)
                                                       (1314)
                                                       (1341)
                                                       (1413)
                                                       (1512)
                                                       (12123)
                                                       (12312)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For leaders of weakly decreasing runs we have A004526.
The case of strict compositions is A096749.
For leaders of anti-runs we have column k = 2 of A374521.
Leaders of strictly increasing runs in standard compositions are A374683.
Ranked by positions of 2s in A374684.
Column k = 2 of A374700.
A003242 counts anti-run compositions.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
Cf. A000009, A096765, A106356, A124766, A238343, A261982, A333213.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#,Less]]==2&]],{n,0,15}]

seq(n)={my(A=O(x^(n-1)), q=eta(x^2 + A)/eta(x + A)); Vec((q*x/(1 + x))^2 + q*x^2/((1 + x)*(1 + x^2)), -n-1)} \\ Andrew Howroyd, Aug 14 2024

G.f.: (x*Q(x)/(1 + x))^2 + x^2*Q(x)/((1 + x)*(1 + x^2)), where Q(x) is the g.f. of A000009. - Andrew Howroyd, Aug 14 2024

a(26) onwards from Andrew Howroyd, Aug 14 2024

A376263 Number of strict integer compositions of n whose leaders of increasing runs are increasing.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 8, 11, 18, 21, 30, 38, 52, 77, 96, 126, 167, 217, 278, 402, 488, 647, 822, 1073, 1340, 1747, 2324, 2890, 3695, 4690, 5924, 7469, 9407, 11718, 15405, 18794, 23777, 29507, 37188, 45720, 57404, 70358, 87596, 110672, 135329, 167018, 206761, 254200, 311920

Offset: 0

Gus Wiseman, Sep 18 2024

nonn

The leaders of increasing runs of a sequence are obtained by splitting it into maximal increasing subsequences and taking the first term of each.

			The a(1) = 1 through a(9) = 11 compositions:
 (1) (2) (3)   (4)   (5)   (6)     (7)     (8)     (9)
         (1,2) (1,3) (1,4) (1,5)   (1,6)   (1,7)   (1,8)
                     (2,3) (2,4)   (2,5)   (2,6)   (2,7)
                           (1,2,3) (3,4)   (3,5)   (3,6)
                           (1,3,2) (1,2,4) (1,2,5) (4,5)
                                   (1,4,2) (1,3,4) (1,2,6)
                                           (1,4,3) (1,3,5)
                                           (1,5,2) (1,5,3)
                                                   (1,6,2)
                                                   (2,3,4)
                                                   (2,4,3)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For less-greater or greater-less we have A294617.
This is a strict case of A374688, weak version A374635.
The strict less-greater version is A374689, weak version A189076.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions, strict A032020.
A238130, A238279, A333755 count compositions by number of runs.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A000110, A008289, A056823, A106356, A188920, A238343, A261982, A274174, A333213, A374634, A374683, A374698, A374763.

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

\\ here Q(n) gives n-th row of A008289.
Q(n)={Vecrev(polcoef(prod(k=1, n, 1 + y*x^k, 1 + O(x*x^n)), n)/y)}
a(n)={if(n==0, 1, my(r=Q(n), s=Vec(serlaplace(exp(exp(x+O(x^#r))- 1)))); sum(k=1, #r, r[k]*s[k]))} \\ Andrew Howroyd, Sep 18 2024

a(n) = Sum_{k>=1} A008289(n,k)*A000110(k-1) for n > 0. - Andrew Howroyd, Sep 18 2024

a(26) onwards from Andrew Howroyd, Sep 18 2024

A331848 Number of partitions of n into odd parts with some part repeated.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 4, 4, 6, 8, 10, 12, 15, 19, 23, 27, 33, 41, 48, 57, 68, 81, 95, 111, 130, 153, 178, 206, 239, 278, 320, 367, 423, 486, 556, 635, 725, 827, 941, 1067, 1211, 1374, 1553, 1753, 1980, 2232, 2512, 2823, 3171, 3560, 3990, 4465, 4995, 5585, 6234, 6951, 7749

Offset: 0

Ilya Gutkovskiy, Jan 29 2020

nonn

Also number of partitions of n into distinct parts with at least one even part.

			a(6) = 3 because we have [3, 3], [3, 1, 1, 1] and [1, 1, 1, 1, 1, 1].

Index entries for sequences related to partitions

Cf. A000009, A000700, A047967, A261982, A331849.

a(n) = A000009(n) - A000700(n).

A331849 Number of partitions of n into prime parts with some part repeated.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 1, 2, 3, 3, 5, 5, 7, 8, 10, 11, 15, 15, 20, 22, 26, 31, 35, 41, 47, 54, 62, 71, 80, 92, 102, 117, 131, 148, 166, 186, 208, 233, 259, 290, 322, 357, 398, 439, 488, 539, 595, 657, 723, 796, 877, 962, 1057, 1159, 1268, 1391, 1520, 1663, 1815, 1981

Offset: 0

Ilya Gutkovskiy, Jan 29 2020

nonn

			a(9) = 3 because we have [5, 2, 2], [3, 3, 3] and [3, 2, 2, 2].

Index entries for sequences related to partitions

Cf. A000586, A000607, A047967, A261982, A331848.

Table[Count[IntegerPartitions[n],?(AllTrue[#,PrimeQ]&&Length[Union[#]]<Length[#]&)],{n,0,60}] (* _Harvey P. Dale, Feb 11 2025 *)

a(n) = A000607(n) - A000586(n).

cp's OEIS Frontend

A374705 Number of integer compositions of n whose leaders of maximal strictly increasing runs sum to 2.

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A376263 Number of strict integer compositions of n whose leaders of increasing runs are increasing.

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A331848 Number of partitions of n into odd parts with some part repeated.

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A331849 Number of partitions of n into prime parts with some part repeated.

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