PROMOTION OF INCREASING TABLEAUX: FRAMES AND
HOMOMESIES
arXiv:1702.01358v1 [math.CO] 5 Feb 2017
OLIVER PECHENIK
Abstract. A key fact about M.-P. Schützenberger’s (1972) promotion operator
on rectangular standard Young tableaux is that iterating promotion once per entry recovers the original tableau. For tableaux with strictly increasing rows and
columns, H. Thomas and A. Yong (2009) introduced a theory of K-jeu de taquin
with applications to K-theoretic Schubert calculus. The author (2014) studied a
K-promotion operator P derived from this theory, but showed that the key fact
does not generally extend to K-promotion of such increasing tableaux.
Here we show that the key fact holds for labels on the boundary of the rectangle.
That is, for T a rectanglar increasing tableau with entries bounded by q, we have
FramepP q pT qq “ FramepT q, where FramepU q denotes the restriction of U to its first
and last row and column. Using this fact, we obtain a family of homomesy results
on the average value of certain statistics over K-promotion orbits, extending a 2-row
theorem of J. Bloom, D. Saracino, and the author (2016) to arbitrary rectangular
shapes.
1. Introduction
An important application of the theory of standard Young tableaux is to the product structure of the cohomology of Grassmannians. Much attention in the modern
Schubert calculus has been devoted to the study of analogous problems in K-theory
(see [PY16, §1] for a partial survey of such work). In particular, H. Thomas and
A. Yong [TY09] gave a K-theoretic Littlewood-Richardson rule by developing a combinatorial theory of increasing tableaux as a K-theoretic analogue of the classical
theory of standard Young tableaux. Their Littlewood-Richardson rule and the associated combinatorics has since been extended to the other minuscule flag varieties
[CTY14, BR12, BS16] and into torus-equivariant K-theory [TY13, PY15].
The theory of increasing tableaux is moreover of independent combinatorial interest. Various enumerative combinatorics results have recently been obtained [Pec14,
PSV16, GMP` 16]; as well as applications to the studies of combinatorial Hopf algebras [PP16], longest increasing subsequences of random words [TY11], plane partitions
[DPS17, HPPW16], and combinatorial representation theory [Rho17]. This paper continues the study begun in [Pec14] of the K-promotion operator on increasing tableaux,
a K-theoretic analogue of M.-P. Schützenberger’s [Sch72] classical promotion operator.
Date: February 7, 2017.
Key words and phrases. Increasing tableau, promotion, K-theory, homomesy, frame.
1
2
O. PECHENIK
We systematically identify a partition λ with its Ferrers diagram in English orientation. An increasing tableau of shape λ is a filling of λ by positive integers such
that entries strictly increase from left to right across rows and from top to bottom
down columns. We write Incq pλq for the set of all increasing tableaux of shape λ
with entries bounded above by q. Using the K-theoretic jeu de taquin of [TY09],
one has a K-promotion operator P on increasing tableaux [Pec14] by analogy with
M.-P. Schützenberger’s classical promotion for standard Young tableaux [Sch72]. We
describe this operator in detail in Section 2.
The operation of (K-)promotion is of particular interest for tableaux of rectangular
shapes. For a standard Young tableau T of shape m ˆ n, one has that P mn pT q “ T
(cf. [Hai92]); indeed, one can completely enumerate the orbits by size in this case via
the cyclic sieving phenomenon [Rho10]. For increasing tableaux, on the other hand,
orbits can be much larger than the cardinality q of the alphabet [Pec14, Example 3.10].
In general, no upper bound is known on the cardinality of the K-promotion orbit of
a increasing tableau, even of rectangular shape.
Example 1.1. Consider the increasing tableau
T “ 1
2
3
4
8 11 13 14 17 P Inc26 p4 ˆ 10q.
7 10 12 13 15 17 19 21
3
6
9 12 13 14 16 18 21 24
6
8 11 15 20 22 23 24 25 26
4
5
7
Although one might naively expect the cardinality of its K-promotion orbit to divide
26 by analogy with the standard Young tableau case, in fact the orbit of T has size
1222 “ 26 ¨ 47.
♦
The frame of a partition λ is the set Framepλq of all boxes in the first or last row,
or in the first or last column. For T P Incq pλq, we write FramepT q for the labeling of
T restricted to Framepλq.
Our first main result is the following:
Theorem 1.2. Let T P Incq pm ˆ nq. Then
FramepT q “ FramepP q pT qq.
Example 1.3. Let T be as in Example 1.1. Then
P 26 pT q “ 1 3 4 5 7 8 11 13 14 17 ,
2 4 6 7 10 12 14 15 19 21
3 6 9 12 13 14 16 18 21 24
6 8 11 15 20 22 23 24 25 26
where we have bolded all entries that coincide with the corresponding entries of T .
Note that in accordance with Theorem 1.2, every entry of FramepP 26 pT qq is bold. ♦
Remark 1.4. Since Framep2 ˆ nq “ 2 ˆ n, Theorem 1.2 in particular recovers the
author’s previous result [Pec14, Theorem 1.3] that P q pT q “ T for T P Incq p2 ˆ nq.
PROMOTION OF INCREASING TABLEAUX
3
The following was conjectured in work with K. Dilks and J. Striker [DPS17, Conjecture 4.12]:
Conjecture 1.5. Let T P Incq p3 ˆ nq. Then T “ P q pT q.
Theorem 1.2 may be interpreted as evidence toward Conjecture 1.5, since Theorem 1.2 shows that T and P q pT q have the same entries in at least 2n ` 2 out of 3n
pairs of corresponding boxes.
A set U of objects with a weight function wt : U Ñ C and a group action
ř G ñ U
wtpxq
is said to be homomesic if every G-orbit O has the same average weight xPO|O| .
This notion was isolated by J. Propp and T. Roby [PR15] in response to observations
of D. Panyushev [Pan09], and has since been found in to appear in diverse situations
[EP14, HZ15, Str15, RW15, DW16, EFG` 16, JR17].
Using Theorem 1.2, we obtain our second main result, a family of new homomesies
for increasing tableaux. For T P Incq pλq and S a set of boxes in λ, let wtS pT q denote
the sum of the entries of T in S.
Theorem 1.6. Let S be subset of Framepm ˆ nq that is fixed under 1800 rotation.
.
Then pIncq pm ˆ nq, P, wtS q exhibits homomesy with orbit average pq`1q|S|
2
Remark 1.7. The case m “ 2 of Theorem 1.6 was previously proved by J. Bloom,
O. Pechenik and D. Saracino [BPS16, Theorem 6.5] using results from [Pec14].
The analogue of Theorem 1.6 for (semi)standard Young tableaux was conjectured
by J. Propp and T. Roby [PR13] and proved by J. Bloom, O. Pechenik and D. Saracino
[BPS16, Theorem 1.1]. In fact, for (semi)standard Young tableaux, S need not be
contained in Framepm ˆ nq. However, [BPS16, Example 6.4] shows that for increasing
tableaux a generalization of Theorem 1.6 without the condition S Ď Framepm ˆ nq
would be false.
2. K-jeu de taquin and frames of increasing tableaux
The section culminates in a proof of Theorem 1.2. First we recall the K-jeu de
taquin of H. Thomas and A. Yong [TY09], the key ingredient in the operation of
K-promotion on increasing tableaux. While K-promotion can be defined without a
full development of K-jeu de taquin, we will need K-jeu de taquin in the proof of
Theorem 1.2.
2.1. K-jeu de taquin. Let BulletTableauxpν{λq denote the set of all fillings of the
skew shape ν{λ by positive integers and symbols ‚. For each positive integer i, we define as follows an operator swapi on BulletTableauxpν{λq. Let T P BulletTableauxpν{λq
and consider the boxes of T that contain either i or ‚. The set of such boxes decomposes into edge-connected components. On each such component that is a single box,
swapi does nothing. On each nontrivial component, swapi simultaneously replaces
each i by ‚ and each ‚ by i. The resulting element of BulletTableauxpν{λq is swapi pT q.
4
O. PECHENIK
Example 2.1. Consider
T “ 4
1
7
3
‚
2
‚
2
‚
3
2
2 P BulletTableaux
ˆ
˙
.
In computing swap2 pT q, one looks at two connected components. The southwest component is a single box containing ‚ and is unchanged by swap2 . The other component
has six boxes. Hence
swap2 pT q “ 4
1
7
3
2
‚
2
‚
‚
3
‚
‚ .
♦
For a box b in a partition, we write bÑ for the box immediately right of b in its
row, bÓ for the box immediately below b in its column, etc.
Consider a skew shape ν{λ. An inner corner of ν{λ is a box b P λ such that bÑ R λ
and bÓ R λ. For I any nonempty set of inner corners of ν{λ and T P Incq pν{λq, let
InI pT q be the extension of T formed by adding a ‚ to each box of I. Note that
InI pT q P BulletTableauxpν{θq for some θ Ą λ.
An outer corner of ν{λ is a box b P ν{λ such that bÑ R ν{λ and bÓ R ν{λ. If
T P BulletTableauxpν{λq has all ‚’s in outer corners, then we define Out‚ pT q to be
the filling obtained by deleting every ‚ from T ; otherwise Out‚ pT q is undefined. Note
that if Out‚ pT q is defined, then it has shape δ{λ for some δ Ď ν.
Let T P Incq pν{λq and let I be any nonempty set of inner corners of ν{λ. Then
the K-jeu de taquin slide of T at I is the result of the following composition of
operations
slideI pT q :“ Out‚ ˝ swapq ˝ ¨ ¨ ¨ ˝ swap2 ˝ swap1 ˝ InI pT q.
Observe that slideI pT q P Incq pδ{ρq for some ρ Ă λ and δ Ă ν.
Iterating this process for successive nonempty sets of inner corners I1 , I2 , . . ., one
eventually obtains an increasing tableau R P Incq pκq of some straight shape κ. Such
a tableau is called a rectification of T .
Remark 2.2. Unlike in the classical standard tableau setting, an increasing tableau
T P Incq pν{λq may have more than one rectification and these rectifications may
moreover have different straight shapes.
2.2. K-promotion. For T P BulletTableauxpν{λq, we define an operation Rep1Ñ‚ that
replaces each instance of 1 by ‚, as well as, for each n P Zą0 , an operation Rep‚Ñn
that replaces each instance of ‚ by n. Let Decr be the operator that decrements each
numerical entry by 1 (and ignores ‚’s).
K-promotion on Incq pλq Ă BulletTableauxpλq is the composition
P :“ Decr ˝ Rep‚Ñq`1 ˝ swapq ˝ ¨ ¨ ¨ ˝ swap3 ˝ swap2 ˝ Rep1Ñ‚ .
PROMOTION OF INCREASING TABLEAUX
5
It is not hard to see that if T P Incq pλq, then PpT q P Incq pλq, and that moreover
this operation coincides with M.-P. Schützenberger’s definition of promotion [Sch72]
in the case that T is a standard Young tableau. For more details, see [Pec14].
Example 2.3. Let T “
1
2
4
3
4
6
P Inc6 p2 ˆ 3q. Then one computes PpT q as
follows:
T “
1
2
4
3
4
6
Rep1Ñ‚
‚
2
4
3
4
6
swap2
2
‚
4
3
4
6
2
4
6
3
6
‚
swap3
2
‚
4
3
4
6
2
4
6
3
6
7
swap4
2
4
‚
3
‚
6
swap5
2
4
‚
3
‚
6
swap6
Rep‚Ñ7
Decr
1
3
5
2
5
6
“ PpT q.
♦
2.3. K-evacuation and its dual. To prove Theorem 1.2 on K-promotion, we will
need the related notion of (dual) K-evacuation. Define the shape of an increasing
tableau T to be shpT q “ λ if T P Incq pλq for some q. Write Tďa for the subtableau
of T given by deleting all entries greater than a and removing all empty boxes. In
analogous fashion, define Tăa , Těa , and Tąa , where Těa and Tąa will generally be of
skew shape. Note that T P Incq pλq is uniquely determined by the vector of partitions
´
¯
shpTď0 q, shpTď1 q, . . . , shpTďq q .
For T P Incq pλq, we define the K-evacuation of T to be the tableau EpT q encoded
by the vector
´
¯
shpP q pT qď0q, shpP q´1 pT qď1 q, . . . , shpP 0 pT qďq q .
Similarly, the dual K-evacuation of T is E ˚pT q encoded by the vector
´
¯
shpP 0 pT qď0q, shpP ´1 pT qď1 q, . . . , shpP ´q pT qďq q .
It is useful to encode all these data in a K-theoretic growth diagram as in
[TY09], using ideas that originate in work of S. Fomin (cf. [Sta99, Appendix 1]); the
K-theoretic growth diagram for T P Incq pλq is a semi-infinite 2-dimensional array
6
O. PECHENIK
formed by placing the partition shpP j pT qďi q in position pi ` j, jq, where 0 ď i ď q
and j P Z.
Example 2.4. Let
T “ 1
3
2
4
4
5
5 .
8
4
6
7
9
6
8 10 11
Then the K-theoretic growth diagram of T is
..
.
H
H
H
H
H
H
H
H
H
H
H
H
..
.
Here the top illustrated row encodes T , the bottom row encodes P 11 pT q, and the
central column encodes EpT q (which is also E ˚ pP 11 pT qq).
♦
Using the K-theoretic growth diagram, it is not hard to uncover various relations between the operators under consideration. Together [TY09, Theorem 4.1] and
[Pec14, Lemma 3.1] give the following facts that we will need:
Lemma 2.5. The following relations hold for operations on Incq pλq:
(a)
(b)
(c)
(d)
E 2 “ pE ˚q2 “ id;
E˚ ˝ E “ Pq;
P ˝ E “ E ˝ P ´1 ;
if λ “ m ˆ n, E ˚ “ rot ˝ E ˝ rot, where rotpT q is given by rotating T by 1800
and replacing i by q ` 1 ´ i.
PROMOTION OF INCREASING TABLEAUX
7
2.4. Proof of Theorem 1.2. Let w be the reading word of T , given by reading
the entries of T by rows from left to right and from bottom to top, i.e. in “reverse
Arabic fashion.” Let rotpwq :“ w0 ¨ w ¨ w0 , where w0 is the longest element of the
symmetric group Sq . Since rotpwq is obtained from w by reversing the order of the
letters of w and then replacing i by q ` 1 ´ i, we see that rotpwq is the reading word
of rotpT q.
Define wďa to be the subword of w obtained by deleting all letters greater than a,
with analogous definitions of wăa , wěa , and wąa .
Lemma 2.6. The tableaux rotpT q and EpT q have the same first row.
Proof. The reading word of rotpT qďa is rotpwqďa. Hence by [TY09, Theorem 6.1], the
length of the first row of rotpT qďa is LISprotpwqďaq, where LISpuq denotes the length
of the longest strictly increasing subsequence of the word u. By the definition of rot,
we have LISprotpwqďa q “ LISpwąn´a q. But by [TY09, Theorem 6.1], LISpwąn´a q is
the length of the first row of any K-rectification of Tąn´a . By definition, the shape
of EpT qďa is the shape of a particular K-rectification of Tąn´a . Thus the length of
the first row of EpT qďa is also the length of the first row of rotpT qďa . The lemma
follows.
Lemma 2.7. The tableaux rotpT q and EpT q have the same first column.
Proof. The proof is the same as for Lemma 2.6, except that one should replace use
of [TY09, Theorem 6.1] on the relation between first rows and longest increasing
subsequences with use of the analogous relation between first columns and longest
decreasing subsequences (see [TY11] or [BS16, Corollary 6.8]).
The following proposition is of independent interest. It extends [Pec14, Proposition 3.3], which is the special case where T P Incq p2 ˆ nq.
Proposition 2.8. The tableaux rotpT q and EpT q have the same frame.
Proof. By Lemmas 2.6 and 2.7, it remains to show that rotpT q and EpT q have the
same last row and column.
Let T 1 “ EpT q. Then by Lemmas 2.6 and 2.7, rotpT 1 q and EpT 1 q have the same first
row and column. But by Lemma 2.5(a), EpT 1 q “ T , so rotpT 1 q and T have the same
first row and column. Hence rotprotpT 1 qq and and rotpT q have the same last row and
column. Since rotprotpT 1 qq “ EpT q, we are done.
Clearly rotpT q and rotprotprotpT qqq have the same frame. Since by Lemma 2.5(d),
we have E ˚ “ rot˝E ˝rot, it thereby follows from Proposition 2.8 that rotpT q and E ˚pT q
also have the same frame. Thus E ˚ pEpT qq has the same frame as rotprotpT qq “ T .
But by Lemma 2.5(b), P q pT q “ E ˚ pEpT qq, so
FramepT q “ FramepP q pT qq.
This concludes the proof of Theorem 1.2.
8
O. PECHENIK
3. Homomesy
In this section, we prove a family of new homomesy results for increasing tableaux.
We will obtain these by imitating the proof of [BPS16, Theorem 1.1] and using Theorem 1.2.
For T P Incq pm ˆ nq and b a box in Framepm ˆ nq, let DistpT, bq be the multiset
DistpT, bq :“ twttbu pP k pT qq : 0 ď k ă qu.
Proposition 3.1. For T P Incq pm ˆ nq and b a box in Framepm ˆ nq,
DistpT, bq “ DistpEpT q, bq.
Proof. This proof is perhaps best understood by following along with the succeeding
Example 3.2.
Consider the K-theoretic growth diagram G for T . A fixed row r of G encodes an
increasing tableau R. The row immediately below this encodes PpRq. The column
that intersects r at its rightmost partition encodes, by definition, EpRq. The column
immediately left of this then encodes PpEpRqq by Lemma 2.5(c). Say the rank of a
partition π in G is the number rankpπq of partitions that are strictly left of π and in
its row. Note that the rank is also the number of partitions strictly below π in its
column.
Shade each partition in G that contains the box b. For any set of q consecutive
rows tri : 0 ă i ď qu, we have by Theorem 1.2 the equality of multisets
DistpT, bq “ trankpρi q : 0 ă i ď qu,
where ρi is the leftmost shaded partition in row ri . In the same way, for any set of q
consecutive columns tcj : 0 ď j ă qu, we have
DistpEpT q, bq “ trankpγj q : 0 ď j ă qu,
where γj is the bottommost shaded partition in column cj .
Fix C P Z. For 1 ď k ď 2q ` 1, let dk denote the diagonal line of slope one through
G given by y “ x´k `C. For each diagonal dk , let δk be the smallest shaded partition
that lies on dk . Observe that the partitions δk are restricted to q ` 1 consecutive rows
tri : 0 ď i ď qu of G and to q ` 1 consecutive columns tcj : 0 ď j ď qu of G. We have
the equalities of multisets
trankpρi q : 0 ă i ď qu “ trankpδk q : 1 ă k ď 2q ` 1, rankpδk q “ rankpδk´1 q ´ 1u
and
trankpγj q : 0 ď j ă qu “ trankpδk q : 1 ď k ă 2q ` 1, rankpδk q “ rankpδk`1 q ´ 1u.
Now construct a lattice path P in the first quadrant of the plane by plotting the
points pk, rankpδk qq for 1 ď k ď 2q ` 1 and connecting the vertex pk, rankpδk qq to the
vertex pk ` 1, rankpδk`1 qq by a line segment. Note that
rankpδ1 q “ rankpδ2q`1 q
by Theorem 1.2. Moreover,
rankpδk`1 q “ rankpδk q ˘ 1
PROMOTION OF INCREASING TABLEAUX
9
for all k. Hence for any positive integer h, the number of k in the interval r2, 2q ` 1s
with rankpδk q “ h “ rankpδk´1 q ´ 1 equals the number of k in the interval r1, 2qs with
rankpδk q “ h “ rankpδk`1 q ´ 1. This proves that
trankpρi q : 0 ă i ď qu “ trankpγj q : 0 ď j ă qu
and the proposition follows.
Example 3.2. Extending Example 2.4, let b be the shaded box in
T “ 1
3
2
4
4
5
5 .
8
4
6
7
9
6
8 10 11
Then we shade the K-theoretic growth diagram of T as
..
.
H
H
H
H
H
d1 d2
H
d3 d4
d5
H
d6
H
d7
d8
H
d9
d10
H
d11 d12
H
d13 d14
H
d15 d16 d17 d18 d19 d20 d21 d22 d23
..
.
10
O. PECHENIK
where we have also labeled 23 consecutive diagonals. We then plot the following
lattice path.
12
10
rankpδk q
8
6
4
2
0
1
3
5
7
9
11
13
15
17
19
21
23
k
Now notice that the number of right-hand endpoints of z steps equals the number of
left-hand endpoints of { steps at each height, the final key observation in the proof of
Proposition 3.1.
♦
3.1. Proof of Theorem 1.6. Consider b P Framepm ˆ nq and let T P Incq pm ˆ nq.
For O the K-promotion orbit of T , we have by Theorem 1.2 that
ř
řq´1
wttbu pP i pT qq
U PO wttbu pUq
“ i“0
.
|O|
q
By Proposition 3.1,
řq´1
i“0
wttbu pP i pT qq
“
q
řq´1
i“0
wttbu pP i pEpT qqq
.
q
But by Theorem 1.2 and Lemma 2.5(c),
řq´1
řq´1
řq´1
i
´i
i
i“0 wttbu pP pEpT qqq
i“0 wttbu pP pEpT qqq
i“0 wttbu pEpP pT qqq
“
“
.
q
q
q
Finally by Proposition 2.8,
˘
řq´1 `
řq´1
řq´1
i
i
i
˚ u pP pT qq
q
`
1
´
wt
wt
protpP
pT
qqq
wt
pEpP
pT
qqq
tb
tbu
tbu
i“0
“ i“0
“ i“0
,
q
q
q
where b˚ is the image of b under rotating m ˆ n by 1800 .
PROMOTION OF INCREASING TABLEAUX
11
Hence putting these facts together, we have
˘ řq´1
řq´1 `
ř
i
wttb˚ u pP i pT qq
i“0 q ` 1 ´ wttb˚ u pP pT qq
U PO wttb,b˚ u pUq
“
` i“0
|O|
q
q
řq´1
pq ` 1q
“ i“0
“ q ` 1.
q
Thus for S any set of boxes in Framepm ˆ nq that is fixed under 1800 rotation, we
have
ř
pq ` 1q|S|
U PO wtS pUq
“
,
|O|
2
as desired.
References
[BPS16]
Jonathan Bloom, Oliver Pechenik, and Dan Saracino, Proofs and generalizations of a
homomesy conjecture of Propp and Roby, Discrete Math. 339 (2016), no. 1, 194–206. 3,
8
[BR12]
Anders Skovsted Buch and Vijay Ravikumar, Pieri rules for the K-theory of cominuscule
Grassmannians, J. Reine Angew. Math. 668 (2012), 109–132. 1
[BS16]
Anders Skovsted Buch and Matthew J. Samuel, K-theory of minuscule varieties, J. Reine
Angew. Math. 719 (2016), 133–171. 1, 7
[CTY14] Edward Clifford, Hugh Thomas, and Alexander Yong, K-theoretic Schubert calculus for
OGpn, 2n ` 1q and jeu de taquin for shifted increasing tableaux, J. Reine Angew. Math.
690 (2014), 51–63. 1
[DPS17]
Kevin Dilks, Oliver Pechenik, and Jessica Striker, Resonance in orbits of plane partitions
and increasing tableaux, J. Combin. Theory Ser. A 148 (2017), 244–274. 1, 3
[DW16]
Chao-Ping Dong and Suijie Wang, Homomesy in the graded poset △p1q, preprint (2016),
arXiv:1606.05715. 3
[EFG` 16] David Einstein, Miriam Farber, Emily Gunawan, Michael Joseph, Matthew Macauley,
James Propp, and Simon Rubinstein-Salzedo, Noncrossing partitions, toggles, and homomesies, Electron. J. Combin. 23 (2016), 1–26. 3
[EP14]
David Einstein and James Propp, Piecewise-linear and birational toggling, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014),
Discrete Math. Theor. Comput. Sci. Proc., AT, Assoc. Discrete Math. Theor. Comput.
Sci., Nancy, 2014, pp. 513–524. 3
[GMP` 16] Christian Gaetz, Michelle Mastrianni, Rebecca Patrias, Hailee Peck, Colleen Robichaux,
David Schwein, and Ka Yu Tam, K-Knuth equivalence for increasing tableaux, Electron.
J. Combin. 23 (2016), no. 1, Paper 1.40, 37. 1
[Hai92]
Mark D. Haiman, Dual equivalence with applications, including a conjecture of Proctor,
Discrete Math. 99 (1992), no. 1-3, 79–113. 2
[HPPW16] Zachary Hamaker, Rebecca Patrias, Oliver Pechenik, and Nathan Williams, Doppelgängers: Bijections of plane partitions, preprint (2016), arXiv:1602.05535. 1
[HZ15]
Sam Hopkins and Ingrid Zhang, A note on statistical averages for oscillating tableaux,
Electron. J. Combin. 22 (2015), no. 2, Paper 2.48, 9. 3
[JR17]
Michael Joseph and Tom Roby, Toggling independent sets of a path graph, preprint
(2017), arXiv:1701.04956. 3
[Pan09]
Dmitri I. Panyushev, On orbits of antichains of positive roots, European J. Combin. 30
(2009), no. 2, 586–594. 3
12
O. PECHENIK
[Pec14]
[PP16]
[PR13]
[PR15]
[PSV16]
[PY15]
[PY16]
[Rho10]
[Rho17]
[RW15]
[Sch72]
[Sta99]
[Str15]
[TY09]
[TY11]
[TY13]
Oliver Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, J. Combin. Theory Ser. A 125 (2014), 357–378. 1, 2, 3, 5, 6, 7
Rebecca Patrias and Pavlo Pylyavskyy, Combinatorics of K-theory via a K-theoretic
Poirier-Reutenauer bialgebra, Discrete Math. 339 (2016), no. 3, 1095–1115. 1
James Propp and Tom Roby, Combinatorial actions & homomesic orbit averages, 23
May 2013, Talk at Dartmouth College Mathematics Colloquium, slides available at
http://www.math.uconn.edu/„troby/homomesyActions2013DartmouthColloq.pdf. 3
, Homomesy in products of two chains, Electron. J. Combin. 22 (2015), no. 3,
Paper 3.4, 29. 3
Timothy Pressey, Anna Stokke, and Terry Visentin, Increasing tableaux, Narayana numbers and an instance of the cyclic sieving phenomenon, Ann. Comb. 20 (2016), no. 3,
609–621. 1
Oliver Pechenik and Alexander Yong, Equivariant K-theory of Grassmannians, preprint
(2015), arXiv:1506.01992. 1
, Genomic tableaux, to appear, J. Algebraic Combin. (2016), arXiv:1603.08490. 1
Brendon Rhoades, Cyclic sieving, promotion, and representation theory, J. Combin.
Theory Ser. A 117 (2010), no. 1, 38–76. 2
, A skein action of the symmetric group on noncrossing partitions, J. Algebraic
Combin. 45 (2017), no. 1, 81–127. 1
David Rush and Kelvin Wang, On orbits of order ideals of minuscule posets II: Homomesy, preprint (2015), arXiv:1509.08047. 3
M. P. Schützenberger, Promotion des morphismes d’ensembles ordonnés, Discrete Math.
2 (1972), 73–94. 1, 2, 5
Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced
Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999, With a foreword
by Gian-Carlo Rota and appendix 1 by Sergey Fomin. 5
Jessica Striker, The toggle group, homomesy, and the Razumov-Stroganov correspondence, Electron. J. Combin. 22 (2015), no. 2, Paper 2.57, 17. 3
Hugh Thomas and Alexander Yong, A jeu de taquin theory for increasing tableaux, with
applications to K-theoretic Schubert calculus, Algebra Number Theory 3 (2009), no. 2,
121–148. 1, 2, 3, 5, 6, 7
, Longest increasing subsequences, Plancherel-type measure and the Hecke insertion algorithm, Adv. in Appl. Math. 46 (2011), no. 1-4, 610–642. 1, 7
, Equivariant schubert calculus and jeu de taquin, to appear, Ann. Inst. Fourier
(Grenoble) (2013). 1
(OP) Department of Mathematics, Rutgers University, Piscataway, NJ 08854
E-mail address: [email protected]