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A090665 Triangle read by rows: T(n,k) = number of preferential arrangements of n things where the first object has rank k. 0
1, 2, 1, 6, 5, 2, 26, 25, 18, 6, 150, 149, 134, 84, 24, 1082, 1081, 1050, 870, 480, 120, 9366, 9365, 9302, 8700, 6600, 3240, 720, 94586, 94585, 94458, 92526, 82320, 57120, 25200, 5040, 1091670, 1091669, 1091414, 1085364, 1038744, 871920, 554400, 221760, 40320 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The rows are the reverses of the rows of A054255.
Row sums give A000670.
Column 1 is A000629. - Joerg Arndt, Dec 08 2014
From Vincent Jackson, May 01 2023: (Start)
The formula
T(n, k) = Sum_{i=k..n-1} i!*StirlingS2(n-1, i) + (k-1)!*StirlingS2(n-1,k-1)
can be derived by splitting the weak orders with the first object at rank k into three categories:
1. weak orders where another object (of the n-1 other objects) has rank k,
2. weak orders where all other objects have rank strictly less than k, and
3. weak orders where no other object is at rank k, but some object has rank greater than k.
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.
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.
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)
LINKS
FORMULA
From Vincent Jackson, May 01 2023: (Start)
T(n, k) = 2*(Sum_{i=k..n-1} i!*StirlingS2(n-1, i)) + (k-1)!*StirlingS2(n-1,k-1).
T(n, k) = 2*A084416(n-1,k) + (k-1)!*StirlingS2(n-1,k-1).
T(n, k) = A084416(n-1,k) + A084416(n-1,k-1). (End)
EXAMPLE
Triangle starts:
01: 1;
02: 2, 1;
03: 6, 5, 2;
04: 26, 25, 18, 6;
05: 150, 149, 134, 84, 24;
06: 1082, 1081, 1050, 870, 480, 120;
07: 9366, 9365, 9302, 8700, 6600, 3240, 720;
08: 94586, 94585, 94458, 92526, 82320, 57120, 25200, 5040;
09: 1091670, 1091669, 1091414, 1085364, 1038744, 871920, 554400, 221760, 40320;
10: 14174522, 14174521, 14174010, 14155350, 13950720, 12930120, 10190880, 5957280, 2177280, 362880;
...
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A228175 A118980 A351385 * A347952 A361198 A359806
KEYWORD
nonn,tabl
AUTHOR
Eugene McDonnell (eemcd(AT)mac.com), Dec 16 2003
EXTENSIONS
Corrected by Alois P. Heinz, Dec 08 2014
Name clarified by Vincent Jackson, May 01 2023
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)