OFFSET
0,5
COMMENTS
T(n,k) = number of operator monomials M in a 5-ary 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) = 21 such operator monomials are L(L(abcde)), L(L(a)bcde), L(L(a))bcde, L(aL(b)cde), L(abL(c)de), L(abcL(d)e), L(abcdL(e)), L(a)L(b)cde, L(a)bL(c)de, L(a)bcL(d)e, L(a)bcdL(e), aL(L(b))cde, aL(b)L(c)de, aL(b)cL(d)e, aL(b)cdL(e), abL(L(c))de, abL(c)L(d)e, abL(c)dL(e), abcL(L(d))e, abcL(d)L(e), abcdL(L(e)).
T(n,k) = number of 231-avoiding permutations of {1,2,...,4n-3k+1} with k+1 decreasing runs, each run having length congruent to 1 modulo 4. E.g., the T(3,2) = 21 such permutations are 7654123, 7653124, 7652134, 7643125, 7642135, 7632145, 7543126, 7542136, 7532146, 7432156, 6543127, 6542137, 6532147, 6432157, 5432167, 1765423, 1765324, 1764325, 1754326, 1654327, 1276543.
T(n,k) = number of labeled Schröder paths of semilength n avoiding DDDDD and having exactly k total occurrences of H and DD where each descent (maximal sequence run of down steps) of length l >=2 is labeled by a (l-1)-subset of {1,2,3}. E.g., the T(3,2) = 21 such paths are
- UDHH, HUDH, HHUD, UHDH, HUHD, UHHD;
- UU(DD)_SH where S belongs to {{1}, {2}, {3}};
- HUU(DD)_S where S belongs to {{1}, {2}, {3}};
- UHU(DD)_S where S belongs to {{1}, {2}, {3}};
- UUH(DD)_S where S belongs to {{1}, {2}, {3}};
- UUU(DDD)_S where S belongs to {{1,2}, {1,3}, {2,3}}.
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>=2 is labeled by a composition of l-1 consisting of 4 nonnegative parts (the final descent is unlabeled). E.g., the T(3,2) = 21 such paths are
- UUUDUDDD, UUDUUDDD, UDUUUDDD;
- UUU(DD)_SUDD where S belongs to {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)};
- UU(DD)_SUUDD where S belongs to {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)};
- UUU(DDD)_SUD where S belongs to {(2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (1,1,0,0), (1,0,1,0), (1,0,0,1), (0,1,1,0), (0,1,0,1), (0,0,1,1)}.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
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.
FORMULA
G.f. u satisfies u = (1+x*y*u)*(1+x*u*(1+x*y*u)^3).
EXAMPLE
Triangle begins:
[0] 1,
[1] 1, 1
[2] 1, 6, 1
[3] 1, 15, 21, 1
[4] 1, 28, 110, 56, 1
[5] 1, 45, 350, 550, 126, 1
[6] 1, 66, 855, 2800, 2145, 252, 1
[7] 1, 91, 1771, 9975, 16660, 7007, 462, 1
[8] 1, 120, 3276, 28336, 83790, 79968, 20020, 792, 1
MATHEMATICA
T[n_, k_]:=Binomial[n+1, k+1]*Binomial[4n-3k+1, k]/(n+1); Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten (* James C. McMahon, Dec 08 2025 *)
PROG
(PARI) T(n, k) = binomial(n+1, k+1) * binomial(4*n-3*k+1, k) / (n+1) \\ Andrew Howroyd, Jan 19 2026
CROSSREFS
KEYWORD
AUTHOR
Yu Hin Au, Nov 27 2025
STATUS
approved