Number of paths from (0,0) to (3n-3,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and have no tripledescents (ddd). Example: a(3)=6 because we have udud, Uddud, udUdd, UddUdd, uudd and Ududd (the remaining four paths contain the string ddd: uUddd, UdUddd, Uuddd and UUdddd; see
A027307). -
Emeric Deutsch, Jun 08 2005
a(n) = number of node-labeled ordered trees (
A000108) on n vertices, each node labeled with a positive integer <= its outdegree. A node is a non-root non-leaf vertex. Example. a(3)=6 counts the 5 ordered trees on 4 vertices with all labels 1 and the tree
.|.
/ \
with its (one and only) node labeled 2. -
David Callan, Jul 14 2006
a(n) = number of Schroeder (n-1)-paths with no triple descents. Example: a(4)=21 counts all 22 Schroeder 3-paths (
A006318) except UUUDDD. -
David Callan, Jul 14 2006
(1 + 2x + 6x^2 + ...)*(1 + x + 2x^2 + 6x^3) = (1 + 3x + 10x^2 + 37x^3 + ...), where
A109081 = (1, 1, 3, 10, 37, ...). -
Gary W. Adamson, Nov 15 2011
a(n) = number of Motzkin paths of length 2n-1 with no downsteps in odd position. Example: a(3)=6 counts FFFFF, FFUDF, FUFDF, UDFFF, UDUDF, UFFDF with U an upstep (1,1), F a flatstep (1,0), and D a downstep (1,-1). -
David Callan, May 20 2015
Number of permutations of length n that avoid 4123, 4132, and 4213. -
Jay Pantone, Oct 01 2015
Conjecturally, the number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) > e(j) and e(i) <= e(k). [Martinez and Savage, 2.21] -
Eric M. Schmidt, Jul 17 2017
a(n) is the number of permutations of length n avoiding the partially ordered pattern (POP) {1>3, 1>4, 4>2} of length 4. That is, the number of length n permutations having no subsequences of length 4 in which the first element is the largest and the fourth element is larger than the second element. -
Sergey Kitaev, Dec 10 2020
a(n) is the number of peakless Motzkin paths of length 2n that do not start with an up edge and where every pair of matching up and down edges occupies positions of the same parity. Equivalently, the number of RNA secondary structures on 2n vertices where the leftmost vertex is not matched and only vertices of the same parity can be matched. -
Alexander Burstein, May 17 2021
a(n) = number of operator monomials M in a ternary associative algebra with a unary linear operator L, where M has n-1 total operations. E.g., the a(4) = 21 such operator monomials are L(L(L(a))), L(L(abc)), L(L(a)bc), L(L(a))bc, L(aL(b)c), L(abL(c)), L(a)L(b)c, L(a)bL(c), aL(L(b))c, aL(b)L(c), abL(L(c)), L(abcde), L(abc)de, L(a)bcde, aL(bcd)e, aL(b)cde, abL(cde), abL(c)de, abcL(d)e, abcdL(e), and abcdefg.
a(n) = number of labeled Dyck paths of semilength n, where every non-terminal descent of length l>=2 is labeled by a composition of l-1 consisting of 2 nonnegative parts (the final descent is unlabeled). E.g., the a(4) = 21 such paths are
- UUUUDDDD, UUUDUDDD, UUDUUDDD, UUDUDUDD, UDUUUDDD, UDUUDUDD, UDUDUUDD, UDUDUDUD;
- UUU(DD)_SUDD, UUDU(DD)_SUD, UU(DD)_SUUDD, UU(DD)_SUDUD, UDUU(DD)_SUD where S belongs to {(1,0), (0,1)};
- UUU(DDD)_SUD where S belongs to {(2,0), (1,1), (0,1)}. (End)
