A173258 -id:A173258 - cp's OEIS frontend

A372647 Number of compositions such that their set adjacent differences are a subset of {-1,1} and contain 1 as a part of the composition itself.

0, 1, 0, 2, 1, 1, 4, 1, 5, 5, 5, 10, 8, 14, 15, 22, 26, 31, 44, 47, 69, 80, 101, 131, 156, 203, 246, 315, 388, 484, 609, 746, 945, 1163, 1453, 1812, 2242, 2799, 3464, 4319, 5351, 6652, 8261, 10235, 12732, 15763, 19577, 24276, 30092, 37338, 46254, 57376, 71081

Offset: 0

John Tyler Rascoe, May 08 2024

nonn

			The compositions for n = 6..8 are:
a(6) = 4: [3,2,1], [2,1,2,1], [1,2,3], [1,2,1,2].
a(7) = 1: [1,2,1,2,1].
a(8) = 5: [2,1,2,1,2], [3,2,1,2], [2,1,2,3], [2,3,2,1], [1,2,3,2].

Cf. (row sums of A372646).

Cf. A003242, A173258, A362500.

A375763 Irregular triangle read by rows, T(n,k) is the number of North-East lattice paths from (0,0) to (n,n+2) that stay weakly above y = x, with weight = k + A000217(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, 4, 5, 4, 4, 3, 2, 1, 1, 1, 4, 7, 10, 11, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1, 1, 5, 11, 18, 24, 27, 30, 29, 28, 25, 23, 19, 16, 12, 10, 7, 5, 3, 2, 1, 1, 1, 6, 16, 30, 46, 59, 71, 78, 81, 81, 78, 74, 67, 60, 52, 46, 37, 31, 24

Offset: 0

John Tyler Rascoe, Aug 26 2024

Here the weight of a lattice path is the area under the path and above the x-axis. T(n,k) also counts the number of integer compositions of (3*n) + (2*k) + 6 with adjacent differences in {-1,1}, first part 1, and last part 3.

			Triangle begins:
    k=0  1  2   3   4   5   6   7   8   9  10  11  12  13  14
 n=0: 1;
 n=1: 1, 1, 1;
 n=2: 1, 2, 2,  2,  1,  1;
 n=3: 1, 3, 4,  5,  4,  4,  3,  2,  1,  1;
 n=4: 1, 4, 7, 10, 11, 11, 11,  9,  8,  6,  5,  3,  2,  1,  1;
 ...
T(1,0) = 1: (NENN).
T(2,1) = 2: (NNEENN) and (NENNEN).
T(3,2) = 4: (NENENNNE), (NENNENEN), (NNEENNEN), and (NNENEENN).

John Tyler Rascoe, Python program.

Cf. A000245 (empirical row sums), A000217 (row lengths).
Cf. A227543 (paths of this kind from (0,0) to (n,n), offset 1 for (0,0) to (n,n+1)).
Cf. A000108, A152659, A173258, A227543, A268429.

Python
```
# see linked program
```

A383620 Number of weak compositions of n such that the set of adjacent differences is a subset of {-1,1}.

Original entry on oeis.org

1, 4, 5, 9, 13, 20, 30, 45, 66, 102, 152, 229, 344, 518, 780, 1180, 1775, 2676, 4037, 6088, 9182, 13852, 20891, 31512, 47536, 71706, 108166, 163172, 246140, 371303, 560118, 844943, 1274606, 1922767, 2900522, 4375493, 6600511, 9956990, 15020307, 22658428

Offset: 0

John Tyler Rascoe, May 02 2025

nonn

			a(0) = 1: (0).
a(1) = 4: (0,1), (0,1,0), (1,0), (1).
...
a(4) = 13: (0,1,0,1,0,1,0,1), (0,1,0,1,0,1,0,1,0), (1,0,1,0,1,0,1,0), (1,0,1,0,1,0,1), (0,1,0,1,2), (1,0,1,2), (2,1,0,1,0), (2,1,0,1), (0,1,2,1,0), (0,1,2,1), (1,2,1,0), (1,2,1), (4).

Alois P. Heinz, Table of n, a(n) for n = 0..5598

Cf. A007318, A173258, A214247, A214249, A227310.

M(k) = matrix(k+1,k+1, i,j, if(i==j,1,if(i==j-1, -x^(i-1), if(i==j+1, -x^(i-1), 0))))
A_x(N) = {my(k=N+1,x='x+O('x^k)); Vec(vecsum(M(k)^(-1) * vector(k+1,i,x^(i-1))~))}
A_x(10)

A383549 Number of rises in all compositions of n with parts in {1,2,3} and adjacent differences in {-1,1}.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 5, 3, 9, 11, 10, 24, 21, 30, 50, 43, 75, 93, 96, 161, 170, 215, 312, 323, 456, 574, 639, 906, 1046, 1276, 1710, 1935, 2501, 3135, 3642, 4760, 5699, 6893, 8823, 10401, 12952, 16079, 19104, 24002, 29097, 35165, 43865, 52628, 64503, 79363, 95329

Offset: 0

John Tyler Rascoe, Apr 29 2025

A rise is any pair of parts (p_{i-1},p_i) with p_{i-1} < p_i.

By reversal a(n) is also the number of descents in all compositions of n of this kind.

			For n = 6 the following compositions have 5 rises: (1,2,1,2), (1,2,3), (2,1,2,1), (3,2,1).

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,2,-1,0,-2,0,-1).

Cf. A011782, A076118, A124760, A151842, A173258, A214247, A220062, A238343.

A_x(N) = {my(x='x+O('x^N)); concat([0,0,0], Vec(x^3*(1 + x^2)^2*(1 + x + x^3)/(1 - x^3 - x^5)^2))}
A_x(40)

G.f.: x^3*(1 + x^2)^2*(1 + x + x^3)/(1 - x^3 - x^5)^2.

cp's OEIS Frontend

A372647 Number of compositions such that their set adjacent differences are a subset of {-1,1} and contain 1 as a part of the composition itself.

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A375763 Irregular triangle read by rows, T(n,k) is the number of North-East lattice paths from (0,0) to (n,n+2) that stay weakly above y = x, with weight = k + A000217(n).

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A383620 Number of weak compositions of n such that the set of adjacent differences is a subset of {-1,1}.

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Programs

PARI

A383549 Number of rises in all compositions of n with parts in {1,2,3} and adjacent differences in {-1,1}.

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PARI

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