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Revision History for A263485 (Bold, blue-underlined text is an addition; faded, red-underlined text is a deletion.)

Showing entries 1-10 | older changes
A263485
Triangle read by rows: T(n,k) (n>=2, 1<=k<=n!) is the number of permutations pi of n such that there are k permutations >= pi in the (strong) Bruhat order.
(history; published version)
#11 by N. J. A. Sloane at Sun Oct 25 15:44:09 EDT 2015
STATUS

editing

approved

#10 by N. J. A. Sloane at Sun Oct 25 15:43:04 EDT 2015
NAME

Triangle read by rows: T(n,k) (n>=2, 1<=k<=n!) is the number of permutations pi of n and having such that there are k permutations that are greater or equal >= pi in the (strong) Bruhat order.

COMMENTS

By symmetry, this is also the number of permutations of n having k the number of permutations pi of n such that there are smaller or equal k permutations <= pi in the (strong) Bruhat order.

STATUS

proposed

editing

Discussion
Sun Oct 25
15:44
N. J. A. Sloane: I reworded the defn and the comment. BTW, in English you have to say "greater than or equal to". But with my change this is irrelevant
#9 by Christian Stump at Sun Oct 25 11:28:40 EDT 2015
STATUS

editing

proposed

#8 by Christian Stump at Sun Oct 25 11:28:32 EDT 2015
COMMENTS

By symmetry, this is also the number of permutations of n having k permutations that are smaller or equal in (strong) Bruhat order.

LINKS

FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000033">The number of permutations greater than or equal to the given permutation in (strong) Bruhat order</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000109">The number of permutations smaller than or equal to the given permutation in (strong) Bruhat order</a>.

STATUS

approved

editing

#7 by Peter Luschny at Sun Oct 25 10:55:13 EDT 2015
STATUS

reviewed

approved

#6 by Peter Luschny at Sun Oct 25 10:53:41 EDT 2015
STATUS

proposed

reviewed

#5 by Christian Stump at Sun Oct 25 10:32:34 EDT 2015
STATUS

editing

proposed

Discussion
Sun Oct 25
10:53
Peter Luschny: It would need another rule for the OEIS-parser, I think. Let's keep things simple and uniform. Thanks.
#4 by Christian Stump at Sun Oct 25 10:32:31 EDT 2015
NAME

Triangle read by rows: T(n,k) (n>=2, 1<=k><=1n!) is the number of permutations of n and having k permutations that are greater or equal in (strong) Bruhat order.

STATUS

proposed

editing

#3 by Christian Stump at Mon Oct 19 14:07:20 EDT 2015
STATUS

editing

proposed

Discussion
Fri Oct 23
17:40
Peter Luschny: The notation T(n>=2, k>=1) is until now not used in the OEIS as far as I know. I personally do not mind it.  What do other editors say?
Sat Oct 24
03:12
Christian Stump: I chose this notation because I found the notion often used, such as by Neil in A262494 seems like too much.
15:28
Peter Luschny: Well, that's different! In this style it would read T(n,k) (n>=2, 1<=k<=n!), right?
16:26
Christian Stump: That's right. It was not that Neil changed my notation, but when I read "T(n,k) (n>=1, 0<=k<n)" I thought that I'd find it much more convenient to only use one bracketing. Anyway, I'd prefer the one I used, but I can as well use the other...
Sun Oct 25
10:10
Peter Luschny: An advantage of  (n>=2, 1<=k<=n!) is that it also gives the upper bound for k (defines the length of the rows).  A second advantage is that it is easier to parse. So yes, I think it is better to use the form T(n,k) (n>=2, 1<=k<=n!).
10:31
Christian Stump: Sure about the upper bound -- I would still prefer T(n>=2, 1<=k<=n!), but anyway...
#2 by Christian Stump at Mon Oct 19 14:06:58 EDT 2015
NAME

allocated for Christian StumpTriangle read by rows: T(n>=2, k>=1) is the number of permutations of n and having k permutations that are greater or equal in (strong) Bruhat order.

DATA

1, 1, 1, 2, 0, 2, 0, 1, 1, 3, 0, 5, 0, 2, 0, 4, 0, 0, 0, 4, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 1, 4, 0, 9, 0, 3, 0, 12, 0, 0, 0, 10, 0, 2, 0, 8, 0, 4, 0, 2, 0, 0, 0, 14, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 1

OFFSET

2,4

COMMENTS

Row sums give A000142, n >= 2.

LINKS

FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000033">The number of permutations greater than or equal to the given permutation in (strong) Bruhat order</a>.

Wikipedia, <a href="https://en.wikipedia.org/wiki/Bruhat_order">Bruhat order</a>

EXAMPLE

Triangle begins:

1,1,

1,2,0,2,0,1,

1,3,0,5,0,2,0,4,0,0,0,4,0,1,0,0,0,2,0,1,0,0,0,1,

...

CROSSREFS

Cf. A000142.

KEYWORD

allocated

nonn,tabf

AUTHOR

Christian Stump, Oct 19 2015

STATUS

approved

editing

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