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A351640 Triangle read by rows: T(n,k) is the number of patterns of length n with all distinct runs and maximum value k.

%I A351640 #10 Jan 28 2023 22:08:07
%S A351640 1,0,1,0,1,2,0,1,4,6,0,1,10,18,24,0,1,16,72,96,120,0,1,34,168,528,600,
%T A351640 720,0,1,52,486,1632,4200,4320,5040,0,1,90,1062,6024,16200,36720,
%U A351640 35280,40320,0,1,152,2460,16896,73200,169920,352800,322560,362880
%N A351640 Triangle read by rows: T(n,k) is the number of patterns of length n with all distinct runs and maximum value k.
%C A351640 A pattern is a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670.
%H A351640 Andrew Howroyd, <a href="/A351640/b351640.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F A351640 T(n,k) = k! * A351641(n,k).
%e A351640 Triangle begins:
%e A351640   1,
%e A351640   0, 1;
%e A351640   0, 1,  2;
%e A351640   0, 1,  4,   6;
%e A351640   0, 1, 10,  18,  24;
%e A351640   0, 1, 16,  72,  96, 120;
%e A351640   0, 1, 34, 168, 528, 600, 720;
%e A351640   ...
%e A351640 The T(3,1) = 1 pattern is 111.
%e A351640 The T(3,2) = 4 patterns are 112, 122, 211, 221.
%e A351640 The T(3,3) = 6 patterns are 123, 132, 213, 231, 312, 321.
%o A351640 (PARI) \\ here LahI is A111596 as row polynomials.
%o A351640 LahI(n, y)={sum(k=1, n, y^k*(-1)^(n-k)*(n!/k!)*binomial(n-1, k-1))}
%o A351640 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 A351640 R(q)={[subst(serlaplace(p), y, 1) | p<-Vec(q)]}
%o A351640 T(n)={my(q=S(n), v=concat([1], sum(k=1, n, R(q^k-1)*sum(r=k, n, y^r*binomial(r, k)*(-1)^(r-k)) ))); [Vecrev(p) | p<-v]}
%o A351640 { my(A=T(10)); for(n=1, #A, print(A[n])) }
%Y A351640 Row sums are A351200.
%Y A351640 Main diagonal is A000142.
%Y A351640 Cf. A000670, A111596, A350824, A351641.
%K A351640 nonn,tabl
%O A351640 0,6
%A A351640 _Andrew Howroyd_, Feb 15 2022