This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A090665 #27 Feb 20 2025 11:20:30
%S A090665 1,2,1,6,5,2,26,25,18,6,150,149,134,84,24,1082,1081,1050,870,480,120,
%T A090665 9366,9365,9302,8700,6600,3240,720,94586,94585,94458,92526,82320,
%U A090665 57120,25200,5040,1091670,1091669,1091414,1085364,1038744,871920,554400,221760,40320
%N A090665 Triangle read by rows: T(n,k) = number of preferential arrangements of n things where the first object has rank k.
%C A090665 The rows are the reverses of the rows of A054255.
%C A090665 Row sums give A000670.
%C A090665 Column 1 is A000629. - _Joerg Arndt_, Dec 08 2014
%C A090665 From _Vincent Jackson_, May 01 2023: (Start)
%C A090665 The formula
%C A090665 T(n, k) = Sum_{i=k..n-1} i!*StirlingS2(n-1, i) + (k-1)!*StirlingS2(n-1,k-1)
%C A090665 can be derived by splitting the weak orders with the first object at rank k into three categories:
%C A090665 1. weak orders where another object (of the n-1 other objects) has rank k,
%C A090665 2. weak orders where all other objects have rank strictly less than k, and
%C A090665 3. weak orders where no other object is at rank k, but some object has rank greater than k.
%C A090665 The number of weak orders in the first category is Sum_{i=k..n-1} i!*StirlingS2(n-1, i), the number of weak orders of length n-1 with number of ranks between k and n-1 (i.e. A084416(n-1,k)). Given a weak order of length n-1 and number of ranks i >= k, the corresponding weak order of length n with the specified object at rank k is formed by inserting the new object into the appropriate rank.
%C A090665 The number of weak orders in the second category is (k-1)!*StirlingS2(n-1,k-1), the number of weak orders of length n-1 with number of ranks k-1. Given a weak order of length n-1 and number of ranks k-1, the corresponding weak order is formed by appending the new object in its own rank.
%C A090665 Lastly, the number of weak orders in the third category is (again) Sum_{i=k..n-1} i!*StirlingS2(n-1, i). Given a weak order of length n-1 and number of ranks k-1, the corresponding weak order is formed by inserting the new object in its own rank after the rank k-1, thereby shifting by one the ranks originally greater than or equal to k. (End)
%F A090665 From _Vincent Jackson_, May 01 2023: (Start)
%F A090665 T(n, k) = 2*(Sum_{i=k..n-1} i!*StirlingS2(n-1, i)) + (k-1)!*StirlingS2(n-1,k-1).
%F A090665 T(n, k) = 2*A084416(n-1,k) + (k-1)!*StirlingS2(n-1,k-1).
%F A090665 T(n, k) = A084416(n-1,k) + A084416(n-1,k-1). (End)
%F A090665 Sum_{k=1..n} k * T(n,k) = A083410(n). - _Alois P. Heinz_, Feb 20 2025
%e A090665 Triangle starts:
%e A090665 01: 1;
%e A090665 02: 2, 1;
%e A090665 03: 6, 5, 2;
%e A090665 04: 26, 25, 18, 6;
%e A090665 05: 150, 149, 134, 84, 24;
%e A090665 06: 1082, 1081, 1050, 870, 480, 120;
%e A090665 07: 9366, 9365, 9302, 8700, 6600, 3240, 720;
%e A090665 08: 94586, 94585, 94458, 92526, 82320, 57120, 25200, 5040;
%e A090665 09: 1091670, 1091669, 1091414, 1085364, 1038744, 871920, 554400, 221760, 40320;
%e A090665 10: 14174522, 14174521, 14174010, 14155350, 13950720, 12930120, 10190880, 5957280, 2177280, 362880;
%e A090665 ...
%t A090665 T = {n, k} |-> 2*Sum[i!*StirlingS2[n-1, i], {i, k, n-1}] + (k-1)i!*StirlingS2[n-1, k-1] (* _Vincent Jackson_, May 01 2023 *)
%Y A090665 Cf. A054255, A008277, A028246, A083410.
%K A090665 nonn,tabl
%O A090665 1,2
%A A090665 Eugene McDonnell (eemcd(AT)mac.com), Dec 16 2003
%E A090665 Corrected by _Alois P. Heinz_, Dec 08 2014
%E A090665 Name clarified by _Vincent Jackson_, May 01 2023