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A351200 Number of patterns of length n with all distinct runs.

%I A351200 #11 Feb 16 2022 16:03:41
%S A351200 1,1,3,11,53,305,2051,15731,135697,1300869,13726431,158137851,
%T A351200 1975599321,26607158781,384347911211,5928465081703,97262304328573,
%U A351200 1691274884085061,31073791192091251,601539400910369671,12238270940611270161,261071590963047040241
%N A351200 Number of patterns of length n with all distinct runs.
%C A351200 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.
%H A351200 Andrew Howroyd, <a href="/A351200/b351200.txt">Table of n, a(n) for n = 0..200</a>
%H A351200 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A351200 The a(1) = 1 through a(3) = 11 patterns:
%e A351200   (1)  (1,1)  (1,1,1)
%e A351200        (1,2)  (1,1,2)
%e A351200        (2,1)  (1,2,2)
%e A351200               (1,2,3)
%e A351200               (1,3,2)
%e A351200               (2,1,1)
%e A351200               (2,1,3)
%e A351200               (2,2,1)
%e A351200               (2,3,1)
%e A351200               (3,1,2)
%e A351200               (3,2,1)
%e A351200 The complement for n = 3 counts the two patterns (1,2,1) and (2,1,2).
%t A351200 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]] /@Subsets[Range[n-1]+1]];
%t A351200 Table[Length[Select[Join@@Permutations/@allnorm[n],UnsameQ@@Split[#]&]],{n,0,6}]
%o A351200 (PARI) \\ here LahI is A111596 as row polynomials.
%o A351200 LahI(n,y)={sum(k=1, n, y^k*(-1)^(n-k)*(n!/k!)*binomial(n-1, k-1))}
%o A351200 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))}
%o A351200 R(q)={[subst(serlaplace(p), y, 1) | p<-Vec(q)]}
%o A351200 seq(n)={my(q=S(n)); concat([1], sum(k=1, n, R(q^k-1)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ _Andrew Howroyd_, Feb 12 2022
%Y A351200 The version for run-lengths instead of runs is A351292.
%Y A351200 A000670 counts patterns, ranked by A333217.
%Y A351200 A005649 counts anti-run patterns, complement A069321.
%Y A351200 A005811 counts runs in binary expansion.
%Y A351200 A032011 counts patterns with distinct multiplicities.
%Y A351200 A044813 lists numbers whose binary expansion has distinct run-lengths.
%Y A351200 A060223 counts Lyndon patterns, necklaces A019536, aperiodic A296975.
%Y A351200 A131689 counts patterns by number of distinct parts.
%Y A351200 A238130 and A238279 count compositions by number of runs.
%Y A351200 A297770 counts distinct runs in binary expansion.
%Y A351200 A345194 counts alternating patterns, up/down A350354.
%Y A351200 Counting words with all distinct runs:
%Y A351200 - A351013 = compositions, for run-lengths A329739, ranked by A351290.
%Y A351200 - A351016 = binary words, for run-lengths A351017.
%Y A351200 - A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
%Y A351200 - A351202 = permutations of prime factors.
%Y A351200 - A351642 = word structures.
%Y A351200 Row sums of A351640.
%Y A351200 Cf. A003242, A098504, A098859, A106356, A242882, A325545, A328592, A329740, A351014, A351204, A351291.
%K A351200 nonn
%O A351200 0,3
%A A351200 _Gus Wiseman_, Feb 09 2022
%E A351200 Terms a(10) and beyond from _Andrew Howroyd_, Feb 12 2022