ABSTRACT
Flattenn(τ) The set of all sps in Pn that avoid the pattern τ Flattenn,k(τ) The set of all sps in Pn,k that avoid the pattern
τ
In set of involutions of [n]∫ A(−→x )dxi The integral of A(−→x ) with respect to xi
∩ Intersection inv(pi) The number of inversions in pi
înv(pi) The number of dual inversions in pi
[k] Set of integers from 1 to k = {1, 2, . . . , k} [k]n Set of words of size n over alphabet [k]
L(pi) The set of leftmost of letters of pi L[[−→x ]] The set of formal power series or generating func-
tions in −→x = (x1, . . . , xk) len(γ) The size of γ lev(pi) Number of levels in pi
Lucn nth Lucas number
M F∼M ′ M and M ′ are Ferrers-equivalent
M F∼M ′ M and M ′ are stack-equivalent
maj(pi) The major index of pi
m̂aj(pi) The dual major index of pi
Motn nth Motzkin number
[n]q The q-number
n! n factorial = n(n− 1) · · · 1 [n]q! The q-analog of the factorial
n!! n double factorial = n(n− 2)(n− 4) · · · NCn The set of noncrossing sps of [n]
nek(pi) The number of k-nestings of pi( n k
) = n!k!(n−k)![
n j
] The q-binomial coefficient
N Set of natural numbers = {1, 2, 3, . . .}
N0 N ∪ {0} N [F (a, a†)] Normal ordering of an operator strongrec Number of strong records in pi
weakrec Number of weak records in pi
addrec Number of additional records in pi occτ (pi) Number occurrences of the pattern τ in pi
: pi : Double dot operation
P(S) The set of all sps of S Pn The set of all sps of [n] pn The number of all sps of [n] Pn,k The set of all sps of [n] with exactly k blocks Pn(τ) The set of all τ -avoiding sps of [n] Pn(T ) The set of all sps of [n] that avoid each pattern
in T Pn,k(τ) The set of all τ -avoiding sps of [n] with exactly
k blocks
Pn,k(T ) The set of all sps of [n] with exactly k blocks that avoid each pattern in T
Pτ (x; q) The gf for the number of sps of [n] according to the number occurrences of pattern τ
Pτ (x, y; q) The gf for the number of sps of [n] with exactly k blocks according to the number occurrences of pattern τ
Pτ (x, y; q|θ1 · · · θm) The gf for the number of sps pi = θ1 · · · θmpi′ of [n] with exactly k blocks according to the number occurrences of pattern τ
Pτ (x; q; k) The gf for the number of sps of [n] with exactly k blocks according to the number occurrences of pattern τ
Pτ (x; q; k|θ1 · · · θm) The gf for the number of sps pi = θ1 · · · θmpi′ of [n] with exactly k blocks according to the number occurrences of pattern τ
R(pi) The set of rightmost of letters of pi R[x] The set of all polynomials in single variable x
Reduce(pi) Reduced form of pi
ris(pi) Number of rises in pi
RL-minima Right-to-left minima
Rn,k The set of all rook placements of n− k rooks on the n-triangular shape
Rowj The set of all cells in row j
Sign(pi) = (−1)n−k The sign of a sp pi of n with exactly k blocks Sn Set of permutations of [n]
ssΛ(M) The number semi-standard fillings of λ that avoid M
σ + k The sequence (σ1 + k) · · · (σn + k), where σ = σ1 · · ·σn
srec(pi) The sum over the positions of all records in pi Stir(n, k) Stirling number of the second kind
S (γ) The set of singletons of γ
T (S) The generating tree whose vertices at level n are the set partitions of Pn(S)
tcross(γ) Total number crossings of γ
τ ∼ ν τ and ν are called Wilf-equivalent τ ∼s ν τ and ν that are strong Wilf-equivalent θp,q Denotes the sequence 2
θp,q Denotes the sequence 1 p21q
Un(t) nth Chebyshev polynomial of the second kind
∪ Union Wq(γ) The q-weight of γ
bxc The largest integer which is small or equal x (z)n Falling polynomial z(z − 1) · · · (z − n+ 1) Z Set of integers = {. . . ,−3,−2,−1, 0, 1, 2, 3, . . .}
