

A351200


Number of patterns of length n with all distinct runs.


25



1, 1, 3, 11, 53, 305, 2051, 15731, 135697, 1300869, 13726431, 158137851, 1975599321, 26607158781, 384347911211, 5928465081703, 97262304328573, 1691274884085061, 31073791192091251, 601539400910369671, 12238270940611270161, 261071590963047040241
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OFFSET

0,3


COMMENTS

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.


LINKS



EXAMPLE

The a(1) = 1 through a(3) = 11 patterns:
(1) (1,1) (1,1,1)
(1,2) (1,1,2)
(2,1) (1,2,2)
(1,2,3)
(1,3,2)
(2,1,1)
(2,1,3)
(2,2,1)
(2,3,1)
(3,1,2)
(3,2,1)
The complement for n = 3 counts the two patterns (1,2,1) and (2,1,2).


MATHEMATICA

allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]] /@Subsets[Range[n1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n], UnsameQ@@Split[#]&]], {n, 0, 6}]


PROG

(PARI) \\ here LahI is A111596 as row polynomials.
LahI(n, y)={sum(k=1, n, y^k*(1)^(nk)*(n!/k!)*binomial(n1, k1))}
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))}
R(q)={[subst(serlaplace(p), y, 1)  p<Vec(q)]}
seq(n)={my(q=S(n)); concat([1], sum(k=1, n, R(q^k1)*sum(r=k, n, binomial(r, k)*(1)^(rk)) ))} \\ Andrew Howroyd, Feb 12 2022


CROSSREFS

The version for runlengths instead of runs is A351292.
A005811 counts runs in binary expansion.
A032011 counts patterns with distinct multiplicities.
A044813 lists numbers whose binary expansion has distinct runlengths.
A131689 counts patterns by number of distinct parts.
A297770 counts distinct runs in binary expansion.
Counting words with all distinct runs:
 A351202 = permutations of prime factors.
Cf. A003242, A098504, A098859, A106356, A242882, A325545, A328592, A329740, A351014, A351204, A351291.


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



