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A351013
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Number of integer compositions of n with all distinct runs.
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36
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1, 1, 2, 4, 7, 14, 26, 48, 88, 161, 294, 512, 970, 1634, 2954, 5156, 9119, 15618, 27354, 46674, 80130, 138078, 232286, 394966, 665552, 1123231, 1869714, 3146410, 5186556, 8620936, 14324366, 23529274, 38564554, 63246744, 103578914, 167860584, 274465845
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OFFSET
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0,3
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LINKS
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EXAMPLE
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The a(1) = 1 through a(5) = 14 compositions:
(1) (2) (3) (4) (5)
(1,1) (1,2) (1,3) (1,4)
(2,1) (2,2) (2,3)
(1,1,1) (3,1) (3,2)
(1,1,2) (4,1)
(2,1,1) (1,1,3)
(1,1,1,1) (1,2,2)
(2,2,1)
(3,1,1)
(1,1,1,2)
(1,1,2,1)
(1,2,1,1)
(2,1,1,1)
(1,1,1,1,1)
For example, the composition c = (3,1,1,1,1,2,1,1,3,4,1,1) has runs (3), (1,1,1,1), (2), (1,1), (3), (4), (1,1), and since (3) and (1,1) both appear twice, c is not counted under a(20).
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], UnsameQ@@Split[#]&]], {n, 0, 10}]
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PROG
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(PARI) \\ here LahI is A111596 as row polynomials.
LahI(n, y) = {sum(k=1, n, y^k*(-1)^(n-k)*(n!/k!)*binomial(n-1, k-1))}
S(n) = {my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); 1 + sum(i=1, (sqrtint(8*n+1)-1)\2, polcoef(p, i, y)*LahI(i, y))}
seq(n)={my(q=S(n)); [subst(serlaplace(p), y, 1) | p<-Vec(prod(k=1, n, subst(q + O(x*x^(n\k)), x, x^k)))]} \\ Andrew Howroyd, Feb 12 2022
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CROSSREFS
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The version for run-lengths instead of runs is A329739, normal A329740.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A116608 counts compositions by number of distinct parts.
A242882 counts compositions with distinct multiplicities.
A297770 counts distinct runs in binary expansion.
A325545 counts compositions with distinct differences.
A329744 counts compositions by runs-resistance.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct runs:
- A351202 = permutations of prime factors.
Cf. A003242, A025047, A044813, A098504, A098859, A106356, A329738, A328592, A334028, A351015, A351201, A351204.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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