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A100630 Array read by antidiagonals: T(m,n) = Sum(1<=i<=m) [ i*(n-1+i)! ].

%I A100630 #15 Jul 22 2025 06:00:16
%S A100630 1,2,5,6,14,23,24,54,86,119,120,264,414,566,719,720,1560,2424,3294,
%T A100630 4166,5039,5040,10800,16680,22584,28494,34406,40319,40320,85680,
%U A100630 131760,177960,224184,270414,316646,362879,362880,766080,1174320,1583280
%N A100630 Array read by antidiagonals: T(m,n) = Sum(1<=i<=m) [ i*(n-1+i)! ].
%C A100630 Inversion vector corresponding to T(m,n): ( n zeros , 1,2,3,...,m , zeros... )
%C A100630 These are the numbers of permutations (in reverse colexicographical order, compare A055089) that reverse a set of consecutive elements and leave all other elements unchanged. Permutation A(m,n) reverses all elements from n to m+n.
%C A100630 The former title of this sequence refers to finite tables of permutations in lexicographical order: "Triangle read by rows: row n gives the index number in the tables of permutations of order n+1, n+2, ... of the permutation in which the first n items are reversed and the remaining items are in order."
%H A100630 Tilman Piesk, <a href="/A100630/b100630.txt">Table of n, a(n) for n = 1..2016</a>
%H A100630 Tilman Piesk, <a href="http://en.wikiversity.org/wiki/Inversion_%28discrete_mathematics%29#arrays3">Inversion (discrete mathematics)</a> (Wikiversity)
%e A100630 T(3,2) = Sum( 1 <= i <= 3 ) [ i * (1+i)! ]
%e A100630 = 1*(1+1)! + 2*(1+2)! + 3*(1+3)!
%e A100630 = 1*2 + 2*6 + 3*24
%e A100630 = 86
%o A100630 MATLAB code is shown on the Wikiversity page.
%o A100630 Using Iverson's J language, A. p for a permutation p gives the row number in the table of permutations of order (length of p) which has p as its value. For example, q 0 4 9 6 7 5 1 11 8 10 2 3 A. q 13610272.
%Y A100630 See A100711 for another version. Row 2 is A052649.
%K A100630 easy,nonn,tabl
%O A100630 1,2
%A A100630 Eugene McDonnell (eemcd(AT)mac.com), Dec 03 2004
%E A100630 Rewritten by _Tilman Piesk_, Jul 13 2012