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%I #23 Sep 28 2020 07:45:17
%S 1,2,3,4,6,8,10,16,22,28,34,58,82,106,130,154,274,394,514,634,754,874,
%T 1594,2314,3034,3754,4474,5194,5914,10954,15994,21034,26074,31114,
%U 36154,41194,46234,86554,126874,167194,207514,247834,288154,328474,368794,409114,771994,1134874,1497754,1860634,2223514,2586394,2949274,3312154,3675034
%N Triangle read by rows, T(n,k) = !n + (k-1)*(n-1)!, with n>=1, 1<=k<=n; Position of the first n-letter permutation beginning with number k in the list of lexicographically sorted permutations A030299.
%C When organized as a triangular table
%C 1;
%C 2, 3;
%C 4, 6, 8;
%C 10, 16, 22, 28;
%C 34, 58, 82, 106, 130;
%C ...
%C the k-th term of row n gives the position of the first n-letter permutation beginning with number k among all the lexicographically ordered permutations A030299. Thus the terms give the positions of rows of irregular table A237265 among the rows of A030298.
%C Note: the notation !n stands for the left factorial, A003422(n).
%H Antti Karttunen, <a href="/A237450/b237450.txt">Rows 1..45 of the triangular table, flattened</a>
%F a(n) = A003422(A002024(n)) + (A002262(n-1)*A000142(A002024(n)-1)).
%t lf[n_] := lf[n] = (-1)^n n! Subfactorial[-n - 1] - Subfactorial[-1] // FullSimplify;
%t T[n_, k_] := lf[n] + (k - 1)(n - 1)!;
%t Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten
%o (Scheme) (define (A237450 n) (+ (A003422 (A002024 n)) (* (A002262 (- n 1)) (A000142 (- (A002024 n) 1)))))
%Y Left edge: A003422.
%Y Cf. also A002024, A002262, A000142, A030298, A030299, A051683, A237265.
%K nonn,tabl
%O 1,2
%A _Antti Karttunen_, Feb 08 2014
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