OFFSET
0,5
COMMENTS
T(n,k) = number of operator monomials M in a ternary associative algebra with a unary linear operator L, where M has n total operations and exactly k occurrences of L. E.g., the T(n,k) = 10 such operator monomials are 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)).
T(n,k) = number of Schröder paths S of semilength 2n-k+1 with k up steps such that S has an odd number of horizontal (H) steps; and between each up step (that goes from y=i to y=i+1) and its matching down step (i.e., the next down step that goes from y=i+1 to y=i), there is an odd number of H steps.
E.g., the T(3,2) = 10 such paths are UUHHHDD, UUHDHHD, UUHDDHH, UHUHDHD, UHHDHDD, UHDUHDH, UHDHUHD, HUUHDDH, HUHDUHD, HHUUHDD.
T(n,k) = number of rooted ordered trees with 2n-k+1 edges and k+1 internal nodes where each internal node has odd outdegree.
T(n,k) = number of Dyck paths of semilength 2n-k+1 with k+1 peaks where every ascent (maximal run of up steps) has odd length. E.g., the T(3,2) = 10 such paths are UUUDUDUDDD, UUUDUDDUDD, UUUDUDDDUD, UUUDDUDUDD, UUUDDUDDUD, UUUDDDUDUD, UDUUUDUDDD, UDUUUDDUDD, UDUUUDDDUD, UDUDUUUDDD.
T(n,k) = number of 231-avoiding permutations of {1,2,...,2n-k+1} with k+1 decreasing runs, each run having odd length. E.g., the T(3,2) = 10 such permutations are 54123, 53124, 43125, 52134, 42135, 32145, 15423, 15324, 14325, 12543.
T(n,k) = number of Schröder paths of semilength n avoiding DDD and having exactly k total occurrences of H and DD. E.g., the T(3,2) = 10 such paths are UDHH, HUDH, HHUD, UUDDH, UHDH, HUUDD, HUHD, UHUDD, UUHDD, UHHD.
T(n,k) = number of labeled lattice paths from (0,0) with n steps of the form (l,1) with integer l>=0 such that the path stays weakly above y=x; every step (l,1) where l >=2 is labeled by a composition of l-1 consisting of 2 nonnegative parts; and exactly k of the n steps have l=0 (i.e., are vertical steps).
T(n,k) = number of labeled Dyck paths of semilength n+1 with k occurrences of UU, where every non-terminal descent of length l is assigned one of l labels (the final descent is unlabeled). E.g., the T(3,2) = 10 such paths are UUU(D)_1UDDD, UUU(DD)_1UDD, UUU(DD)_2UDD, UUU(DDD)_1UD, UUU(DDD)_2UD, UUU(DDD)_3UD, UU(D)_1UUDDD, UU(DD)_1UUDD, UU(DD)_2UUDD, U(D)_1UUUDDD.
T(n,k) = number of labeled rooted ordered trees with n+1 edges and k+1 leaves where each non-root internal node with outdegree l is assigned one of l labels.
LINKS
Yu Hin Au and Murray R. Bremner, A new generalization of the Narayana numbers inspired by linear operators on associative d-ary algebras, arXiv:2511.13671 [math.CO], 2025.
David Callan and Toufik Mansour, Enumeration of small Wilf classes avoiding 1342 and two other 4-letter patterns, Pure Mathematics and Applications (2018) Vol. 27, No. 1, 62-97. See Section 3.16.
JiSun Huh, Sangwook Kim, Seunghyun Seo, and Heesung Shin, Bijections on pattern avoiding inversion sequences and related objects, Advances in Applied Mathematics (2024) Vol. 161, Paper No. 102771.
FORMULA
G.f. u satisfies u = (1+x*y*u)*(1+x*u*(1+x*y*u)).
EXAMPLE
The initial rows of the triangle are:
[0] 1
[1] 1, 1
[2] 1, 4, 1
[3] 1, 9, 10, 1
[4] 1, 16, 42, 20, 1
[5] 1, 25, 120, 140, 35, 1
[6] 1, 36, 275, 600, 378, 56, 1
[7] 1, 49, 546, 1925, 2310, 882, 84, 1
[8] 1, 64, 980, 5096, 10010, 7392, 1848, 120, 1
MATHEMATICA
T[n_, k_]:=Binomial[n+1, k+1]*Binomial[2n-k+1, k]/(n+1); Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* James C. McMahon, Dec 04 2025 *)
CROSSREFS
KEYWORD
AUTHOR
Yu Hin Au, Nov 26 2025
STATUS
approved