We study the gaps Δp_n between consecutive rank sizes in r-differential posets by introducing a projection operator whose matrix entries can be expressed in terms of the number of certain paths in the Hasse diagram. We strengthen Miller’s result that Δp_n ≥ 1, which resolved a longstanding conjecture of Stanley, by showing that Δp_n ≥ 2r. We also obtain stronger bounds in the case that the poset has many substructures called threads

Gaetz, Christian

Venkataramana, Praveen

English

Caltech Authors - Main

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PATH COUNTING AND RANK GAPS IN DIFFERENTIAL
POSETS
CHRISTIAN GAETZ
Department of Mathematics, Massachusetts Institute of Technology,
Cambridge, MA.
PRAVEEN VENKATARAMANA
Division of Physics, Mathematics, and Astronomy, California Institute of
Technology, Pasadena, CA.
Abstract. We study the gaps ∆pn between consecutive rank sizes in
r-differential posets by introducing a projection operator whose matrix
entries can be expressed in terms of the number of certain paths in the
Hasse diagram. We strengthen Miller’s result that ∆pn ≥ 1, which
resolved a longstanding conjecture of Stanley, by showing that ∆pn ≥
2r. We also obtain stronger bounds in the case that the poset has many
substructures called threads.
1. Introduction
Differential posets are a class of partially ordered sets, originally defined
by Stanley [6], which generalize many of the enumerative and combinatorial
properties of Young’s lattice Y , the lattice of integer partitions. We refer
the reader to [7] for definitions and basic facts about posets.
If P is a graded poset and S ⊆ P , let QS denote the Q-vector space with
basis S. Let Pn denote the n-th rank of P and define the up and down
E-mail addresses: gaetz@mit.edu, pvenkata@caltech.edu.
Date: January 6, 2020.
Key words and phrases. Differential poset; rank growth; thread element; Hasse dia-
gram; path counting.
C.G. was supported by the National Science Foundation Graduate Research Fellowship
under Grant No. 1122374.
1
2 PATH COUNTING AND RANK GAPS IN DIFFERENTIAL POSETS
Figure 1. The Hasse diagram of the Fibonacci lattice Z(2),
a 2-differential poset, up to rank 3. The thread elements (see
Section 3) are unfilled.
operators Un : QPn → QPn+1 and Dn : QPn → QPn−1 by
Unx =
∑
x⋖y
y
Dnx =
∑
z⋖x
z
where x ∈ Pn and ⋖ denotes the covering relation in P . We often omit
the subscripts and write U,D when no confusion can result. For r ∈ Z>0
a locally-finite N-graded poset P with 0̂ is an r-differential poset if, for all
n ≥ 0 we have
Dn+1Un − Un−1Dn = r · I(1)
as linear operators.
Example 1.1. Young’s lattice Y is a 1-differential poset (see Figure 2).
More generally the product poset Y r is an r-differential poset.
Another family of examples are the Fibonacci lattices Z(r) (see Figure
1). They are produced by the following iterative procedure:
(1) Having defined a differential poset P up to rank n, add an element
x̂ to Pn+1 for each element x ∈ Pn−1 so that x̂ covers exactly those
y ∈ Pn which cover x.
(2) For each y ∈ Pn, add an additional r singletons covering y.
The r-differential poset Z(r) is obtained by iterating this procedure begin-
ning with a single point [6].
One of the most basic properties of differential posets that one might hope
to study are the possible rank functions pn = |Pn|, not least because the rank
functions for the examples Y and Z(1) from Example 1.1 are the integer
partition numbers and Fibonacci numbers respectively, two sequences of
PATH COUNTING AND RANK GAPS IN DIFFERENTIAL POSETS 3
Figure 2. The Hasse diagram of Young’s lattice Y , a 1-
differential poset, up to rank 5. The thread elements (see
Section 3) are unfilled.
integers which have received an immense amount of study. The first general
result obtained about pn was the following:
Proposition 1.2 (Stanley [6]). For any differential poset P the rank sizes
p0, p1, p2, ... weakly increase.
Two main lines of research about the rank function have developed, both
motivated by the intuition that Y r should be the “smallest” r-differential
poset, and Z(r) the “largest”.
The first concerns the asymptotic behavior of pn. Stanley and Zanello [8]
have shown that in an r-differential poset,
pn ≫ na · e
√
2rn
a bound which is fairly close to the well-known asymptotics for pr(n) =
|(Y r)n|:
pr(n) ≫ c · nb · eπ
√
2rn/3
where a, b, c are constants. On the other hand, Byrnes ([2], Theorem 1.2)
has shown that for any r-differential poset
pn ≤ |Z(r)n|.
From the recursive definition of Z(r) in Example 1.1 it is immediate that
|Z(r)|n = r · |Z(r)|n−1 + |Z(r)|n−2. Standard techniques allow us to solve
4 PATH COUNTING AND RANK GAPS IN DIFFERENTIAL POSETS
this recurrence and observe that
|Z(r)|n =
r +
√
r2 + 4
2
√
r2 + 4
· ϕnr +
√
r2 + 4− r
2
√
r2 + 4
· ψnr
≤ 2 · ϕnr(2)
where ϕr = (r +
√
r2 + 4)/2 and ψr = (r −
√
r2 + 4)/2.
The second, more difficult, line of research has sought explicit bounds on
the rank gaps ∆pn = pn − pn−1, which, by Proposition 1.2 are nonnegative.
These rank gaps are the eigenvalue multiplicities of DUn (see [6]) and also
appear in the study of the finer algebraic structure of differential posets (see
[1, 4]). It was conjectured in 1988 in Stanley’s original paper [6] that, except
when r = n = 1, the rank gaps ∆pn are strictly positive. This conjecture
was proven by Miller only as recently as 2013, who showed:
Theorem 1.3 (Miller [5]). For any r-differential poset, ∆pn ≥ 1, unless
r = n = 1.
Despite the fact that Stanley’s conjecture was open for so long, it seems
that much stronger results may be true. For example, Stanley and Zanello
[8] have asked whether all iterated partial differences ∆tpn are eventually
positive for large n, and whether this is already true for n = 2 in the case
t = r. One implication of a positive answer to this question would be that
(3) lim
r→∞
min
P r-differential
n≥2
∆pn = ∞
and that
(4) lim
n→∞
∆pn = ∞
for any differential poset P .
Our main theorem is the first improvement on Miller’s bound for general
differential posets, and resolves (3). In addition, it seems plausible that the
new methods we use to prove this result might lead to a resolution of (4),
see Remark 2.
Theorem 1.4. For any r-differential poset, ∆pn ≥ 2r for n ≥ 4 if r = 1,
n ≥ 3 if r = 2 and for n ≥ 2 if r > 2.
In Theorem 3.3 we also prove a stronger result in the case where P has
many threads (see Section 3).
2. Projection matrices and path counting
Let 〈, 〉 denote the inner product on QP which makes the elements of
P into an orthonormal basis. It is easy to see that the operators U,D are
adjoint with respect to this inner product:
〈Ux, y〉 = 〈x,Dy〉.
PATH COUNTING AND RANK GAPS IN DIFFERENTIAL POSETS 5
Formulas for counting paths in the Hasse diagram of P (viewed as a graph)
can often be expressed in terms of the inner product. For example e(x) :=
〈Un0̂, x〉 for x ∈ Pn is easily seen to be the number of paths in the Hasse
diagram of P from 0̂ to x using only upward steps. We will make use of the
following enumerative identity:
Proposition 2.1 (Stanley [6]). Let P be an r-differential poset, then
∑
x∈Pn
e(x)2 = rnn!.
The following linear algebraic facts will also be used:
Proposition 2.2 (Stanley [6]). Let P be a differential poset and n ≥ 0,
then:
(a) Un : QPn → QPn+1 is injective, and
(b) DUn is invertible.
Our main tool will be to study the projection operator onto im(U), whose
entries can be expressed in terms of path counting in the Hasse diagram of
P .
Proposition 2.3. Let Mn = U(DU)
−1Dn, then:
(a) M is the orthogonal projection QPn ։ im(Un−1).
(b) In the standard basis Pn for QPn, the entries for Mn = (mxy)x,y∈Pn
are given by
mxy =
n∑
k=1
(−1)k−1 〈D
kx,Dky〉
rkk!
.
Futhermore 〈Dkx,Dky〉 counts the number of pairs of paths of length
k beginning at x and y and ending at a common element of Pn−k.
Proof. We have
M2 = U(DU)−1DU(DU)−1D =M,
so M is a projection. Since D and (DU)−1 are surjective by Proposition 2.2
(D is the transpose of an injective map), im(Mn) = im(Un−1), so M is a
projection onto im(Un−1). Orthogonality follows easily from the adjointness
of U,D, so (a) is proven. For (b), we first show that
(DUn−1)
−1 =
n−1∑
k=0
(−1)k U
kDk
rk+1(k + 1)!
.
6 PATH COUNTING AND RANK GAPS IN DIFFERENTIAL POSETS
Indeed, let R denote the right-hand side, then:
DUn−1 ·R =
n−1∑
k=0
(−1)k DUU
kDk
rk+1(k + 1)!
=
n−1∑
k=0
(−1)kU
k+1Dk+1 + r(k + 1)UkDk
rk+1(k + 1)!
= I + (−1)n−1U
nDn
rnn!
= I.
where the second equality follows from applying (1), the third from collaps-
ing the telescoping sum, and the last from observing that Dn = 0 on QPn−1.
Now, by definition we have
Mn =
n−1∑
k=0
(−1)k U
k+1Dk+1
rk+1(k + 1)!
.
Reindexing and taking matrix entries by applying adjointness, we obtain the
desired result. 
Remark 1. The formula for (DUn+1)
−1, which plays a key role in the proof of
Proposition 2.3, was suggested to us in personal communication by Alexan-
der Miller, who more generally gave a formula for (UDn+1 + kI)
−1 (our
formula is the case where k = r). The expression for mxy is also similar to
an expression obtained by Miller in [5] for some of the entries in (DU+kI)−1.
Motivated by an earlier representation-theoretic proof of strict rank growth
in the case of differential towers of groups [4] (see also [3]), Miller studied the
denominators in (DU + kI)−1, rather than the eigenvalues of submatrices
of M , as we do.
We will always think of Mn as a matrix in the standard basis for QPn.
Given S ⊆ Pn, let MS denote the principal submatrix of Mn with rows and
columns indexed by the elements of S.
Lemma 2.4. Let S ⊆ Pn and suppose MS has no eigenvalue equal to 1,
then ∆pn ≥ |S|.
Proof. IfMS has no eigenvalue equal to 1, then QS must intersect im(Un−1)
trivially. Since Un−1 is injective, this means that
dim(QPn) ≥ dim(QPn−1) + dim(QS).
And so ∆pn ≥ |S|. 
3. Thread elements and rank gaps
It seems very difficult to estimate the entries mxy for arbitrary x, y ∈
Pn using the formula in Proposition 2.3, since the sum is alternating and
individual terms can be much larger than the sum (it follows from the fact
PATH COUNTING AND RANK GAPS IN DIFFERENTIAL POSETS 7
that M2 = M that |mxy| ≤ 1, however individual terms in the sum can be
much larger than 1). We therefore define special elements for which mxy
can be more easily analyzed.
An element x ∈ P is called a singleton if x = 0̂ or x covers a unique
element of P . A thread element is a singleton x ∈ P which covers another
singleton. A key property of thread elements is the following:
Proposition 3.1 (Miller and Reiner [4]). Let t0 ∈ P be a thread element.
Then there exist thread elements
t0 ⋖ t1 ⋖ t2 ⋖ · · ·
We call this infinite sequence a thread.
Given a differential poset P , let τn denote the number of thread elements
in Pn. Proposition 3.1 implies that τ is a nondecreasing function of n.
Example 3.2. If P = Y r, then the thread elements in ranks n ≥ 2 are the
elements (∅, ..., ∅, λ, ∅, ..., ∅) where λ = (n) or (1n). Thus τn = 2r for n ≥ 2.
If P = Z(r), we get r thread elements in rank n for each singleton in
Pn−1, and there are r of these singletons for each element of Pn−2. Thus
τn = r
2pn−2; in particular, τ grows exponentially in n by (2).
Theorem 3.3. Let P be an r-differential poset. Then for N ≥ 4n:
∆pN ≥ τn.
Proof. Assume n ≥ 1, since τ0 = 0. Let T ⊂ Pn be the set of thread elements
in Pn, so |T | = τn. Suppose N ≥ 4n and for each element t ∈ T , extend t
to a thread via Proposition 3.1, and let t̂ be the element of this thread at
rank N . It is clear from the definition of threads that ŝ 6= t̂ for s 6= t. Let
T̂ = {t̂ | t ∈ T}. We now want to show that M
T̂
has no eigenvalue equal to
1, and apply Lemma 2.4.
Let s, t ∈ T . We first bound the tail of the sum formula for mŝt̂ given in
Proposition 2.3. We have
∣∣∣∣∣
N∑
k=N−n+1
(−1)k−1 〈D
kŝ, Dk t̂〉
rkk!
∣∣∣∣∣ =
∣∣∣∣∣
N∑
k=N−n+1
(−1)k−1 〈D
k−N+ns,Dk−N+nt〉
rkk!
∣∣∣∣∣
(5)
≤
N∑
k=N−n+1
rnn!
rkk!
(6)
≤ n · r
nn!
r3n(3n)!
.(7)
The equality (5) follows because ŝ, t̂ are thread elements, so DN−nŝ = s and
DN−nt̂ = t. The inequality (6) uses Proposition 2.1: the number of paths
down from s to 0̂ is at most
√
rnn!, and thus the number of paths down
from s and ending in any particular rank is at most this number; as this
8 PATH COUNTING AND RANK GAPS IN DIFFERENTIAL POSETS
also holds for t, the bound follows. Letting a(r, n) denote the expression on
the right-hand side of (7), all off-diagonal entries of MT̂ satisfy
|mŝt̂| ≤ a(r, n),
since ŝ and t̂ lie on separate threads and have no common lower bounds
above rank n. The diagonal entries satisfy
mt̂t̂ ≤
(
1
r
− 1
2r2
+ · · · ± 1
rN−n(N − n)!
)
+ a(r, n)
<
(
1
r
− 1
2r2
+
1
6r3
)
+ a(r, n)
≤ 2
3
+ a(r, n).
Let ϕr =
1
2
(r +
√
r2 + 4). By Byrnes’ result (2), pn ≤ 2ϕnr , and so T̂ has
at most this size. By Gershgorin’s Circle Theorem, in order to see that M
T̂
has no eigenvalue equal to 1, it suffices to show that
2
3
+ 2ϕnr a(r, n) ≤ 1.
If r > 1, then r2n > ϕnr and 2nn!/(3n)! ≤ 1/3. If r = 1, we have τ1 = 1, so
Theorem 1.3 gives the result when n = 1; for n ≥ 2 it is easy to see that
2ϕn1a(1, n) < 1/3. 
We now give the proof of Theorem 1.4.
Proposition 3.4. Let P be an r-differential poset. Then at least two thread
elements cover each x ∈ P1.
Proof. In any r-differential poset P , Equation (1) implies that there are r
elements in P1, and all of these are singletons, so they must each be covered
by r+1 elements. Given x ∈ P1, for each y in P1 different from x, there is at
most one element covering both x, y, as can easily be seen from the defining
relation (1). Thus there are at most r−1 elements covering x which are not
singletons, and so at least two thread elements cover x. 
Proof of Theorem 1.4. This proof is similar to the proof of Theorem 3.3,
except that we can be more precise, since the threads involved begin in
ranks 0 or 1. We prove the case r ≥ 3 here, the other cases are nearly
identical, except that the different values of r in the denominators imply
that larger ranks n must be considered in order for the Gershgorin bound
to hold.
Let r ≥ 3, n ≥ 2 and let T = {t1, ..., tr, s1, ..., sr} ⊂ P2 be a set of 2r
thread elements, with ti, si covering each element xi of P1; such elements
exist by Proposition 3.4. Extend each t ∈ T to an infinite thread, and let
T̂ = {t̂ | t ∈ T} be the elements of rank n in these threads. Then M
T̂
has
PATH COUNTING AND RANK GAPS IN DIFFERENTIAL POSETS 9
diagonal elements:
mt̂t̂ =
n−1∑
k=0
(−1)k 〈D
k+1t̂, Dk+1t̂〉
rk+1(k + 1)!
=
1
r
− 1
2!r2
+ · · ·+ (−1)n−1 1
n!rn
<
1
r
≤ 1
3
for some k. For off-diagonal entries we have
|mt̂i,ŝi| =
∣∣∣∣
1
rn−1(n− 1)! −
1
rnn!
∣∣∣∣ ≤
1
rn−1(n− 1)! ≤
1
3
,
and all other off-diagonal entries satisfy
|mŝt̂| =
∣∣∣∣
1
rnn!
∣∣∣∣ ≤
1
2r2
.
The sum of the absolute values of the matrix entries of M
T̂
across a row is
thus at most 1/3 + 1/3 + (2r − 2)/2r2 < 1. Therefore by the Gershgorin
Circle Theorem, MT̂ has no eigenvalue equal to 1. Applying Lemma 2.4
completes the proof.

Remark 2. One can obtain stronger bounds in an ad-hoc manner using
Lemma 2.4. For example, it is possible to show that when r = 1 we have
∆pn ≥ 3 for n ≥ 6 by letting sn, tn be the elements of rank n in the
two threads which begin in rank 2, and letting un be a certain sequence
of elements which are “close” to sn so that the entries of M{sn,tn,un} can
be well approximated by exhaustively considering all possible paths. What
is needed to progress further on this problem is identify families Sn ⊂ Pn
whose size grows with n for which the entries of MSn can be estimated.
Conjecture 3.5. In any differential poset limn→∞∆pn = ∞.
Acknowledgements
The authors wish to thank Fabrizio Zanello for helping to initiate this
joint project and Richard Stanley for his helpful conversations. We are
also grateful to Patrick Byrnes for making his computer code available and
Alexander Miller for providing useful references.
References
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groups. Algebr. Comb., 2(6):1311–1327, 2019.
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MI, 2012. Thesis (Ph.D.)–University of Minnesota.
[3] Christian Gaetz. Dual graded graphs and Bratteli diagrams of towers of groups. Elec-
tron. J. Combin., 26(1):Paper 1.25, 12, 2019.
[4] Alexander Miller and Victor Reiner. Differential posets and Smith normal forms. Order,
26(3):197–228, 2009.
[5] Alexander R. Miller. Differential posets have strict rank growth: a conjecture of Stan-
ley. Order, 30(2):657–662, 2013.
[6] Richard P. Stanley. Differential posets. J. Amer. Math. Soc., 1(4):919–961, 1988.
10 PATH COUNTING AND RANK GAPS IN DIFFERENTIAL POSETS
[7] Richard P. Stanley. Enumerative combinatorics. Volume 1, volume 49 of Cambridge
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[8] Richard P. Stanley and Fabrizio Zanello. On the rank function of a differential poset.
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Path Counting and Rank Gaps in Differential Posets

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