A072704 -id:A072704 - cp's OEIS frontend

A293136 Irregular triangle T(n,k) read by rows: T(n,k) is the number of strongly unimodal compositions of n (A059618) into k parts.

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 4, 1, 0, 1, 4, 5, 0, 1, 6, 6, 2, 0, 1, 6, 10, 4, 0, 1, 8, 14, 6, 1, 0, 1, 8, 19, 14, 1, 0, 1, 10, 23, 20, 5, 0, 1, 10, 31, 30, 10, 0, 1, 12, 36, 42, 18, 2, 0, 1, 12, 44, 60, 27, 4, 0, 1, 14, 52, 76, 48, 8, 0, 1, 14, 61, 102, 68, 16, 1, 0, 1, 16, 69, 126, 101, 30, 1, 0, 1, 16, 81, 160, 138, 50, 5, 0

Offset: 0

Joerg Arndt, Oct 01 2017

Conjecture: index k of last nonzero entry in row n of is A293137(n).

			Triangle starts:
00:  [1]
01:  [0, 1]
02:  [0, 1]
03:  [0, 1, 2]
04:  [0, 1, 2, 1]
05:  [0, 1, 4, 1]
06:  [0, 1, 4, 5]
07:  [0, 1, 6, 6, 2]
08:  [0, 1, 6, 10, 4]
09:  [0, 1, 8, 14, 6, 1]
10:  [0, 1, 8, 19, 14, 1]
11:  [0, 1, 10, 23, 20, 5]
12:  [0, 1, 10, 31, 30, 10]
13:  [0, 1, 12, 36, 42, 18, 2]
14:  [0, 1, 12, 44, 60, 27, 4]
15:  [0, 1, 14, 52, 76, 48, 8]
16:  [0, 1, 14, 61, 102, 68, 16, 1]
17:  [0, 1, 16, 69, 126, 101, 30, 1]
18:  [0, 1, 16, 81, 160, 138, 50, 5]
19:  [0, 1, 18, 90, 194, 191, 80, 10]
20:  [0, 1, 18, 102, 238, 252, 118, 22]
...
Row n=7 is [0, 1, 6, 6, 2] because in the 15 partitions of 7 there is 0 into zero parts, 1 into one part, 6 into two parts, 6 into three parts, and 2 into four parts:
[ 1]   [ 1 2 3 1 ]
[ 2]   [ 1 2 4 ]
[ 3]   [ 1 3 2 1 ]
[ 4]   [ 1 4 2 ]
[ 5]   [ 1 5 1 ]
[ 6]   [ 1 6 ]
[ 7]   [ 2 3 2 ]
[ 8]   [ 2 4 1 ]
[ 9]   [ 2 5 ]
[10]   [ 3 4 ]
[11]   [ 4 2 1 ]
[12]   [ 4 3 ]
[13]   [ 5 2 ]
[14]   [ 6 1 ]
[15]   [ 7 ]

Joerg Arndt, Table of n, a(n) for n = 0..1793 (rows 0...125)

Cf. A059618 (row sums), A293137.

Cf. A072704 (same for weakly unimodal compositions).

N=25;  x='x+O('x^N);
T=Vec(1 + sum(n=1, N, t*x^(n) * prod(k=1, n-1, 1+t*x^k)^2));
for(r=1,#T, print(Vecrev(T[r])) );  \\ as triangle

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

A332871 Number of compositions of n whose run-lengths are not weakly increasing.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 8, 24, 55, 128, 282, 625, 1336, 2855, 6000, 12551, 26022, 53744, 110361, 225914, 460756, 937413, 1902370, 3853445, 7791647, 15732468, 31725191, 63907437, 128613224, 258626480, 519700800, 1043690354, 2094882574, 4202903667, 8428794336, 16897836060

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

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

Also compositions whose run-lengths are not weakly decreasing.

			The a(4) = 1 through a(6) = 8 compositions:
  (112)  (113)   (114)
         (221)   (1113)
         (1112)  (1131)
         (1121)  (1221)
                 (2112)
                 (11112)
                 (11121)
                 (11211)
For example, the composition (2,1,1,2) has run-lengths (1,2,1), which are not weakly increasing, so (2,1,1,2) is counted under a(6).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for the compositions themselves (not run-lengths) is A056823.
The version for unsorted prime signature is A112769, with dual A071365.
The case without weakly decreasing run-lengths either is A332833.
The complement is counted by A332836.
Compositions that are not unimodal are A115981.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are not unimodal are A332727.
Cf. A001523, A072704, A100883, A181819, A329744, A329766, A332641, A332669, A332726, A332745, A332746, A332834, A332835.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!LessEqual@@Length/@Split[#]&]],{n,0,10}]

a(n) = 2^(n - 1) - A332836(n).

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

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A293136 Irregular triangle T(n,k) read by rows: T(n,k) is the number of strongly unimodal compositions of n (A059618) into k parts.

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