|
| |
|
|
A348013
|
|
Triangle by rows: T(n,k) is the number of n-step Dyck paths with k catastrophes.
|
|
1
|
|
|
|
1, 1, 1, 3, 2, 1, 4, 7, 3, 1, 10, 14, 12, 4, 1, 15, 37, 31, 18, 5, 1, 35, 74, 90, 56, 25, 6, 1, 56, 176, 216, 179, 90, 33, 7, 1, 126, 352, 552, 492, 315, 134, 42, 8, 1, 210, 794, 1269, 1362, 966, 510, 189, 52, 9, 1, 462, 1588, 3033, 3480, 2890, 1716, 777, 256, 63, 10, 1, 792, 3473
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,4
|
|
|
COMMENTS
|
T(n,k) is the number chains of k "incomplete" Dyck paths with a total length of n. (Incomplete Dyck paths are those not ending at the horizontal axis.) Each of the k subsections of the paths does not return to the horizontal axis; they are commonly referred to as paths with catastrophes (like black Fridays on stock market charts).
|
|
|
LINKS
|
|
|
|
FORMULA
|
The generating function of column k is g037952(x)^k, where g037952(x) = x +x^2 +3*x^3+... is the generating function of A037952.
|
|
|
EXAMPLE
|
The triangle starts
1
1 1
3 2 1
4 7 3 1
10 14 12 4 1
15 37 31 18 5 1
35 74 90 56 25 6 1
56 176 216 179 90 33 7 1
126 352 552 492 315 134 42 8 1
210 794 1269 1362 966 510 189 52 9 1
462 1588 3033 3480 2890 1716 777 256 63 10 1
792 3473 6781 8901 8060 5521 2835 1130 336 75 11 1
T(1,1)=1 counts U| where the vertical bar indicates starting a new path at the horizontal axis (the catastrophe).
T(2,1)=1 counts UU|.
T(4,1)=4 counts UUUU|, UUUD|, UUDU|, UDUU|.
T(3,2)=2 counts UU|U| and U|UU| .
T(4,2)=7 counts U|UUU|, U|UUD|, U|UDU|, UU|UU|, UUU|U|, UUD|U| and UDU|U|.
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
