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Plane partition
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<h2 id="generating-function-of-plane-partitions">Generating function of plane partitions</h2> <p>The <a href="/facts/Generating_function/K7UVjnjL">generating function</a> for PL(<i>n</i>) is<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> ∑ n = 0 ∞ PL ( n ) x n = ∏ k = 1 ∞ 1 ( 1 − x k ) k = 1 + x + 3 x 2 + 6 x 3 + 13 x 4 + 24 x 5 + ⋯ {\displaystyle \sum _{n=0}^{\infty }\operatorname {PL} (n)x^{n}=\prod _{k=1}^{\infty }{\frac {1}{(1-x^{k})^{k}}}=1+x+3x^{2}+6x^{3}+13x^{4}+24x^{5}+\cdots \qquad } (sequence A000219 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>). <p>It is sometimes referred to as the <i>MacMahon function</i>, as it was discovered by <a href="/facts/Percy_Alexander_MacMahon/scUegUSn">Percy A. MacMahon</a>. </p><p>This formula may be viewed as the 2-dimensional analogue of <a href="/facts/Leonhard_Euler/z2fsbjgj">Euler</a>'s <a href="/facts/Integer_partition/LtyXWqwH">product formula</a> for the number of <a href="/facts/Partition_(number_theory)/LtyXWqwH">integer partitions</a> of <i>n</i>. There is no analogous formula known for partitions in higher dimensions (i.e., for <a href="/facts/Solid_partition/CbBnHIH5">solid partitions</a>).<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> The asymptotics for plane partitions were first calculated by <a href="/facts/E._M._Wright/SS01ywc2">E. M. Wright</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> One obtains, for large n {\displaystyle n} , that<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> PL ( n ) ∼ ζ ( 3 ) 7 / 36 12 π ( n 2 ) − 25 / 36 exp ( 3 ζ ( 3 ) 1 / 3 ( n 2 ) 2 / 3 + ζ ′ ( − 1 ) ) . {\displaystyle \operatorname {PL} (n)\sim {\frac {\zeta (3)^{7/36}}{\sqrt {12\pi }}}\ \left({\frac {n}{2}}\right)^{-25/36}\ \exp \left(3\ \zeta (3)^{1/3}\left({\frac {n}{2}}\right)^{2/3}+\zeta '(-1)\right).} <p>Evaluating numerically yields </p> ln PL ( n ) ∼ 2.00945 n 2 / 3 − 0.69444 ln n − 1.4631. {\displaystyle \ln \operatorname {PL} (n)\sim 2.00945n^{2/3}-0.69444\ln n-1.4631.} <h3>Plane partitions in a box</h3> <p>Around 1896, MacMahon set up the generating function of plane partitions that are subsets of the r × s × t {\displaystyle r\times s\times t} box B ( r , s , t ) = { ( i , j , k ) | 1 ≤ i ≤ r , 1 ≤ j ≤ s , 1 ≤ k ≤ t } {\displaystyle {\mathcal {B}}(r,s,t)=\{(i,j,k)|1\leq i\leq r,1\leq j\leq s,1\leq k\leq t\}} in his first paper on plane partitions.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> The formula is given by ∑ π ∈ B ( r , s , t ) q | π | = ∏ i = 1 r ∏ j = 1 s 1 − q i + j + t − 1 1 − q i + j − 1 {\displaystyle \sum _{\pi \in {\mathcal {B}}(r,s,t)}q^{|\pi |}=\prod _{i=1}^{r}\prod _{j=1}^{s}{\frac {1-q^{i+j+t-1}}{1-q^{i+j-1}}}} </p><p>A proof of this formula can be found in the book <i>Combinatory Analysis</i> written by MacMahon.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> MacMahon also mentions the generating functions of plane partitions.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> The formula for the generating function can be written in an alternative way, which is given by ∑ π ∈ B ( r , s , t ) q | π | = ∏ i = 1 r ∏ j = 1 s ∏ k = 1 t 1 − q i + j + k − 1 1 − q i + j + k − 2 {\displaystyle \sum _{\pi \in {\mathcal {B}}(r,s,t)}q^{|\pi |}=\prod _{i=1}^{r}\prod _{j=1}^{s}\prod _{k=1}^{t}{\frac {1-q^{i+j+k-1}}{1-q^{i+j+k-2}}}} </p><p>Multiplying each component by 1 − q 1 − q {\displaystyle \textstyle {\frac {1-q}{1-q}}} , and setting <i>q</i> = 1 in the formulas above yields that the total number N 1 ( r , s , t ) {\displaystyle N_{1}(r,s,t)} of plane partitions that fit in the r × s × t {\displaystyle r\times s\times t} box B ( r , s , t ) {\displaystyle {\mathcal {B}}(r,s,t)} is equal to the following product formula:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> N 1 ( r , s , t ) = ∏ ( i , j , k ) ∈ B ( r , s , t ) i + j + k − 1 i + j + k − 2 = ∏ i = 1 r ∏ j = 1 s i + j + t − 1 i + j − 1 . {\displaystyle N_{1}(r,s,t)=\prod _{(i,j,k)\in {\mathcal {B}}(r,s,t)}{\frac {i+j+k-1}{i+j+k-2}}=\prod _{i=1}^{r}\prod _{j=1}^{s}{\frac {i+j+t-1}{i+j-1}}.} The planar case (when <i>t</i> = 1) yields the <a href="/facts/Binomial_coefficient/Tsx7eY5h">binomial coefficients</a>: </p> B ( r , s , 1 ) = ( r + s r ) . {\displaystyle {\mathcal {B}}(r,s,1)={\binom {r+s}{r}}.} <p>The general solution is </p> B ( r , s , t ) = ∏ k = 1 t ( r + s + k − 1 ) ! ( k − 1 ) ! ( r + k − 1 ) ! ( s + k − 1 ) ! = ∏ k = 1 t ( r + s + k − 1 r + k − 1 ) ( r + s + k − 1 s + k − 1 ) ( r + s + k − 1 r ) ( s + k − 1 s ) {\displaystyle {\begin{aligned}{\mathcal {B}}(r,s,t)&=\prod _{k=1}^{t}{\frac {(r+s+k-1)!(k-1)!}{(r+k-1)!(s+k-1)!}}\\&=\prod _{k=1}^{t}{\frac {{\binom {r+s+k-1}{r+k-1}}{\binom {r+s+k-1}{s+k-1}}}{{\binom {r+s+k-1}{r}}{\binom {s+k-1}{s}}}}\end{aligned}}} <p>The <a href="/facts/Isometric_projection/9pf2WSC0">isometric projection</a> of the unit cubes representing a plane partition in a box gives a bijection between these plane partitions and rhombus tilings of a hexagon with the same edge lengths as the box.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> <h2 id="special-plane-partitions">Special plane partitions</h2> <p>Special plane partitions include symmetric, cyclic and self-complementary plane partitions, and combinations of these properties. </p><p>In the subsequent sections, the enumeration of special sub-classes of plane partitions inside a box are considered. These articles use the notation N i ( r , s , t ) {\displaystyle N_{i}(r,s,t)} for the number of such plane partitions, where r, s, and t are the dimensions of the box under consideration, and i is the index for the case being considered. </p> <h3>Action of <i>S</i>2, <i>S</i>3 and <i>C</i>3 on plane partitions</h3> <p> S 2 {\displaystyle {\mathcal {S}}_{2}} is the group of <a href="/facts/Symmetric_group/RUksK5l5">permutations</a> acting on the first two coordinates of a point. This group contains the identity, which sends (<i>i</i>, <i>j</i>, <i>k</i>) to itself, and the transposition (<i>i</i>, <i>j</i>, <i>k</i>) → (<i>j</i>, <i>i</i>, <i>k</i>). The number of elements in an orbit η {\displaystyle \eta } is denoted by | η | {\displaystyle |\eta |} . B / S 2 {\displaystyle {\mathcal {B}}/{\mathcal {S}}_{2}} denotes the set of orbits of elements of B {\displaystyle {\mathcal {B}}} under the action of S 2 {\displaystyle {\mathcal {S}}_{2}} . The height of an element (<i>i</i>, <i>j</i>, <i>k</i>) is defined by h t ( i , j , k ) = i + j + k − 2. {\displaystyle ht(i,j,k)=i+j+k-2.} The height increases by one for each step away from the back right corner. For example, the corner position (1, 1, 1) has height 1 and <i>ht</i>(2, 1, 1) = 2. The height of an orbit is defined to be the height of any element in the orbit. This notation of the height differs from the notation of <a href="/facts/Ian_G._Macdonald/PoaeNbeg">Ian G. Macdonald</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p><p>There is a natural action of the permutation group S 3 {\displaystyle {\mathcal {S}}_{3}} on a Ferrers diagram of a plane partition—this corresponds to simultaneously permuting the three coordinates of all nodes. This generalizes the conjugation operation for <a href="/facts/Integer_partition/LtyXWqwH">integer partitions</a>. The action of S 3 {\displaystyle {\mathcal {S}}_{3}} can generate new plane partitions starting from a given plane partition. Below there are shown six plane partitions of 4 that are generated by the S 3 {\displaystyle {\mathcal {S}}_{3}} action. Only the exchange of the first two coordinates is manifest in the representation given below. </p> 3 1 3 1 2 1 1 2 1 1 1 1 1 1 1 1 1 1 {\displaystyle {\begin{smallmatrix}3&1\end{smallmatrix}}\quad {\begin{smallmatrix}3\\1\end{smallmatrix}}\quad {\begin{smallmatrix}2&1&1\end{smallmatrix}}\quad {\begin{smallmatrix}2\\1\\1\end{smallmatrix}}\quad {\begin{smallmatrix}1&1&1\\1\end{smallmatrix}}\quad {\begin{smallmatrix}1&1\\1\\1\end{smallmatrix}}} <p> C 3 {\displaystyle {\mathcal {C}}_{3}} is called the group of cyclic permutations and consists of </p> ( i , j , k ) → ( i , j , k ) , ( i , j , k ) → ( j , k , i ) , and ( i , j , k ) → ( k , i , j ) . {\displaystyle (i,j,k)\rightarrow (i,j,k),\quad (i,j,k)\rightarrow (j,k,i),\quad {\text{and }}\quad (i,j,k)\rightarrow (k,i,j).} <h3>Symmetric plane partitions</h3> <p>A plane partition π {\displaystyle \pi } is called symmetric if π<i>i</i>,<i>j</i> = π<i>j,i</i> for all <i>i</i>, <i>j</i>. In other words, a plane partition is symmetric if ( i , j , k ) ∈ B ( r , s , t ) {\displaystyle (i,j,k)\in {\mathcal {B}}(r,s,t)} if and only if ( j , i , k ) ∈ B ( r , s , t ) {\displaystyle (j,i,k)\in {\mathcal {B}}(r,s,t)} . Plane partitions of this type are symmetric with respect to the plane <i>x</i> = <i>y</i>. Below is an example of a symmetric plane partition and its visualisation. </p> 4 3 3 2 1 3 3 2 1 3 2 2 1 2 1 1 1 {\displaystyle {\begin{matrix}4&3&3&2&1\\3&3&2&1&\\3&2&2&1&\\2&1&1&&\\1&&&&\end{matrix}}} <p>In 1898, MacMahon formulated his conjecture about the generating function for symmetric plane partitions which are subsets of B ( r , r , t ) {\displaystyle {\mathcal {B}}(r,r,t)} .<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> This conjecture is called The MacMahon conjecture. The generating function is given by ∑ π ∈ B ( r , r , t ) / S 2 q | π | = ∏ i = 1 r [ 1 − q t + 2 i − 1 1 − q 2 i − 1 ∏ j = i + 1 r 1 − q 2 ( i + j + t − 1 ) 1 − q 2 ( i + j − 1 ) ] {\displaystyle \sum _{\pi \in {\mathcal {B}}(r,r,t)/{\mathcal {S}}_{2}}q^{|\pi |}=\prod _{i=1}^{r}\left[{\frac {1-q^{t+2i-1}}{1-q^{2i-1}}}\prod _{j=i+1}^{r}{\frac {1-q^{2(i+j+t-1)}}{1-q^{2(i+j-1)}}}\right]} </p><p>Macdonald<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> pointed out that Percy A. MacMahon's conjecture reduces to </p> ∑ π ∈ B ( r , r , t ) / S 2 q | π | = ∏ η ∈ B ( r , r , t ) / S 2 1 − q | η | ( 1 + h t ( η ) ) 1 − q | η | h t ( η ) {\displaystyle \sum _{\pi \in {\mathcal {B}}(r,r,t)/{\mathcal {S}}_{2}}q^{|\pi |}=\prod _{\eta \in {\mathcal {B}}(r,r,t)/{\mathcal {S}}_{2}}{\frac {1-q^{|\eta |(1+ht(\eta ))}}{1-q^{|\eta |ht(\eta )}}}} <p>In 1972 Edward A. Bender and <a href="/facts/Donald_Knuth/GozNzDM6">Donald E. Knuth</a> conjectured<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> a simple closed form for the generating function for plane partition which have at most <i>r</i> rows and strict decrease along the rows. <a href="/facts/George_Andrews_(mathematician)/x58GmSuR">George Andrews</a> showed<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> that the conjecture of Bender and Knuth and the MacMahon conjecture are equivalent. MacMahon's conjecture was proven almost simultaneously by George Andrews in 1977<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> and later Ian G. Macdonald presented an alternative proof.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> When setting <i>q</i> = 1 yields the counting function N 2 ( r , r , t ) {\displaystyle N_{2}(r,r,t)} which is given by </p> N 2 ( r , r , t ) = ∏ i = 1 r 2 i + t − 1 2 i − 1 ∏ 1 ≤ i < j ≤ r i + j + t − 1 i + j − 1 {\displaystyle N_{2}(r,r,t)=\prod _{i=1}^{r}{\frac {2i+t-1}{2i-1}}\prod _{1\leq i<j\leq r}{\frac {i+j+t-1}{i+j-1}}} <p>For a proof of the case <i>q</i> = 1 please refer to George Andrews' paper <i>MacMahon's conjecture on symmetric plane partitions</i>.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p> <h3>Cyclically symmetric plane partitions</h3> <p>π is called cyclically symmetric, if the <i>i</i>-th row of π {\displaystyle \pi } is conjugate to the <i>i</i>-th column for all <i>i</i>. The <i>i</i>-th row is regarded as an ordinary partition. The conjugate of a partition π {\displaystyle \pi } is the partition whose diagram is the transpose of partition π {\displaystyle \pi } .<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> In other words, the plane partition is cyclically symmetric if whenever ( i , j , k ) ∈ B ( r , s , t ) {\displaystyle (i,j,k)\in {\mathcal {B}}(r,s,t)} then (<i>k</i>, <i>i</i>, <i>j</i>) and (<i>j</i>, <i>k</i>, <i>i</i>) also belong to B ( r , s , t ) {\displaystyle {\mathcal {B}}(r,s,t)} . Below an example of a cyclically symmetric plane partition and its visualization is given. </p> 6 5 5 4 3 3 6 4 3 3 1 6 4 3 1 1 4 2 2 1 3 1 1 1 1 1 {\displaystyle {\begin{matrix}6&5&5&4&3&3\\6&4&3&3&1&\\6&4&3&1&1&\\4&2&2&1&&\\3&1&1&&&\\1&1&1&&&\end{matrix}}} <p>Macdonald's conjecture provides a formula for calculating the number of cyclically symmetric plane partitions for a given integer <i>r</i>. This conjecture is called The Macdonald conjecture. The generating function for cyclically symmetric plane partitions which are subsets of B ( r , r , r ) {\displaystyle {\mathcal {B}}(r,r,r)} is given by </p> ∑ π ∈ B ( r , r , r ) / C 3 q | π | = ∏ η ∈ B ( r , r , r ) / C 3 1 − q | η | ( 1 + h t ( η ) ) 1 − q | η | h t ( η ) {\displaystyle \sum _{\pi \in {\mathcal {B}}(r,r,r)/{\mathcal {C}}_{3}}q^{|\pi |}=\prod _{\eta \in {\mathcal {B}}(r,r,r)/{\mathcal {C}}_{3}}{\frac {1-q^{|\eta |(1+ht(\eta ))}}{1-q^{|\eta |ht(\eta )}}}} <p>This equation can also be written in another way </p> ∏ η ∈ B ( r , r , r ) / C 3 1 − q | η | ( 1 + h t ( η ) ) 1 − q | η | h t ( η ) = ∏ i = 1 r [ 1 − q 3 i − 1 1 − q 3 i − 2 ∏ j = i r 1 − q 3 ( r + i + j − 1 ) 1 − q 3 ( 2 i + j − 1 ) ] {\displaystyle \prod _{\eta \in {\mathcal {B}}(r,r,r)/{\mathcal {C}}_{3}}{\frac {1-q^{|\eta |(1+ht(\eta ))}}{1-q^{|\eta |ht(\eta )}}}=\prod _{i=1}^{r}\left[{\frac {1-q^{3i-1}}{1-q^{3i-2}}}\prod _{j=i}^{r}{\frac {1-q^{3(r+i+j-1)}}{1-q^{3(2i+j-1)}}}\right]} <p>In 1979, Andrews proved Macdonald's conjecture for the case <i>q</i> = 1 as the "weak" Macdonald conjecture.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> Three years later William H. Mills, <a href="/facts/David_P._Robbins/Uy3wE7EQ">David Robbins</a> and Howard Rumsey proved the general case of Macdonald's conjecture in their paper <i>Proof of the Macdonald conjecture</i>.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> The formula for N 3 ( r , r , r ) {\displaystyle N_{3}(r,r,r)} is given by the "weak" Macdonald conjecture </p> N 3 ( r , r , r ) = ∏ i = 1 r [ 3 i − 1 3 i − 2 ∏ j = i r i + j + r − 1 2 i + j − 1 ] {\displaystyle N_{3}(r,r,r)=\prod _{i=1}^{r}\left[{\frac {3i-1}{3i-2}}\prod _{j=i}^{r}{\frac {i+j+r-1}{2i+j-1}}\right]} <h3>Totally symmetric plane partitions</h3> <p>A totally symmetric plane partition π {\displaystyle \pi } is a plane partition which is symmetric and cyclically symmetric. This means that the diagram is symmetric at all three diagonal planes, or in other words that if ( i , j , k ) ∈ B ( r , s , t ) {\displaystyle (i,j,k)\in {\mathcal {B}}(r,s,t)} then all six permutations of (<i>i</i>, <i>j</i>, <i>k</i>) are also in B ( r , s , t ) {\displaystyle {\mathcal {B}}(r,s,t)} . Below an example of a matrix for a totally symmetric plane partition is given. The picture shows the visualisation of the matrix. </p> 5 4 4 3 1 4 3 3 1 4 3 2 1 3 1 1 1 {\displaystyle {\begin{matrix}5&4&4&3&1\\4&3&3&1&\\4&3&2&1&\\3&1&1&&\\1&&&&\end{matrix}}} <p>Macdonald found the total number of totally symmetric plane partitions that are subsets of B ( r , r , r ) {\displaystyle {\mathcal {B}}(r,r,r)} . The formula is given by </p> N 4 ( r , r , r ) = ∏ η ∈ B ( r , r , r ) / S 3 1 + h t ( η ) h t ( η ) {\displaystyle N_{4}(r,r,r)=\prod _{\eta \in {\mathcal {B}}(r,r,r)/{\mathcal {S}}_{3}}{\frac {1+ht(\eta )}{ht(\eta )}}} <p>In 1995 <a href="/facts/John_Stembridge/mhpk0oo1">John R. Stembridge</a> first proved the formula for N 4 ( r , r , r ) {\displaystyle N_{4}(r,r,r)} <a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> and later in 2005 it was proven by George Andrews, <a href="/facts/Peter_Paule/hwybeCqC">Peter Paule</a>, and Carsten Schneider.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> Around 1983 Andrews and Robbins independently stated an explicit product formula for the orbit-counting generating function for totally symmetric plane partitions.<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a><a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> This formula already alluded to in George E. Andrews' paper <i>Totally symmetric plane partitions</i> which was published 1980.<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> The conjecture is called The <i>q</i>-TSPP conjecture and it is given by: </p><p>Let S 3 {\displaystyle {\mathcal {S}}_{3}} be the symmetric group. The orbit counting function for totally symmetric plane partitions that fit inside B ( r , r , r ) {\displaystyle {\mathcal {B}}(r,r,r)} is given by the formula </p> ∑ π ∈ B ( r , r , r ) / S 3 q | π | = ∏ η ∈ B ( r , r , r ) / S 3 1 − q 1 + h t ( η ) 1 − q h t ( η ) = ∏ 1 ≤ i ≤ j ≤ k ≤ r 1 − q i + j + k − 1 1 − q i + j + k − 2 . {\displaystyle \sum _{\pi \in {\mathcal {B}}(r,r,r)/{\mathcal {S}}_{3}}q^{|\pi |}=\prod _{\eta \in {\mathcal {B}}(r,r,r)/{\mathcal {S}}_{3}}{\frac {1-q^{1+ht(\eta )}}{1-q^{ht(\eta )}}}=\prod _{1\leq i\leq j\leq k\leq r}{\frac {1-q^{i+j+k-1}}{1-q^{i+j+k-2}}}.} <p>This conjecture was proved in 2011 by <a href="/facts/Christoph_Koutschan/r3G4VFCQ">Christoph Koutschan</a>, <a href="/facts/Manuel_Kauers/ffIQvP7N">Manuel Kauers</a> and <a href="/facts/Doron_Zeilberger/M90TmhyY">Doron Zeilberger</a>.<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a> </p> <h3>Self-complementary plane partitions</h3> <p>If π i , j + π r − i + 1 , s − j + 1 = t {\displaystyle \pi _{i,j}+\pi _{r-i+1,s-j+1}=t} for all 1 ≤ i ≤ r {\displaystyle 1\leq i\leq r} , 1 ≤ j ≤ s {\displaystyle 1\leq j\leq s} , then the plane partition is called self-complementary. It is necessary that the product r ⋅ s ⋅ t {\displaystyle r\cdot s\cdot t} is even. Below an example of a self-complementary symmetric plane partition and its visualisation is given. </p> 4 4 3 2 1 4 2 2 2 3 2 1 {\displaystyle {\begin{matrix}4&4&3&2&1\\4&2&2&2&\\3&2&1&&\end{matrix}}} <p><a href="/facts/Richard_P._Stanley/8bQ4QrLM">Richard P. Stanley</a><a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a> conjectured formulas for the total number of self-complementary plane partitions N 5 ( r , s , t ) {\displaystyle N_{5}(r,s,t)} . According to Stanley, Robbins also formulated formulas for the total number of self-complementary plane partitions in a different but equivalent form. The total number of self-complementary plane partitions that are subsets of B ( r , s , t ) {\displaystyle {\mathcal {B}}(r,s,t)} is given by </p> N 5 ( 2 r , 2 s , 2 t ) = N 1 ( r , s , t ) 2 {\displaystyle N_{5}(2r,2s,2t)=N_{1}(r,s,t)^{2}} N 5 ( 2 r + 1 , 2 s , 2 t ) = N 1 ( r , s , t ) N 1 ( r + 1 , s , t ) {\displaystyle N_{5}(2r+1,2s,2t)=N_{1}(r,s,t)N_{1}(r+1,s,t)} N 5 ( 2 r + 1 , 2 s + 1 , 2 t ) = N 1 ( r + 1 , s , t ) N 1 ( r , s + 1 , t ) {\displaystyle N_{5}(2r+1,2s+1,2t)=N_{1}(r+1,s,t)N_{1}(r,s+1,t)} <p>It is necessary that the product of <i>r,s</i> and <i>t</i> is even. A proof can be found in the paper <i>Symmetries of Plane Partitions</i> which was written by Stanley.<a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a><a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> The proof works with Schur functions s s r ( x ) {\displaystyle s_{s^{r}}(x)} . Stanley's proof of the ordinary enumeration of self-complementary plane partitions yields the <i>q</i>-analogue by substituting x i = q i {\displaystyle x_{i}=q^{i}} for i = 1 , … , n {\displaystyle i=1,\ldots ,n} .<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a> This is a special case of Stanley's hook-content formula.<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a> The generating function for self-complementary plane partitions is given by </p> s γ α ( q , q 2 , … , q n ) = q γ α ( α + 1 ) / 2 ∏ i = 1 α ∏ j = 0 γ − 1 1 − q i + n − α + j 1 − q i + j {\displaystyle s_{\gamma ^{\alpha }}(q,q^{2},\ldots ,q^{n})=q^{\gamma \alpha (\alpha +1)/2}\prod _{i=1}^{\alpha }\prod _{j=0}^{\gamma -1}{\frac {1-q^{i+n-\alpha +j}}{1-q^{i+j}}}} <p>Substituting this formula in </p> s s r ( x 1 , x 2 , … , x t + r ) 2 for B ( 2 r , 2 s , 2 t ) {\displaystyle s_{s^{r}}(x_{1},x_{2},\ldots ,x_{t+r})^{2}\ {\text{ for }}{\mathcal {B}}(2r,2s,2t)} s s r ( x 1 , x 2 , … , x t + r ) s ( s + 1 ) r ( x 1 , x 2 , … , x t + r ) for B ( 2 r , 2 s + 1 , 2 t ) {\displaystyle s_{s^{r}}(x_{1},x_{2},\ldots ,x_{t+r})s_{(s+1)^{r}}(x_{1},x_{2},\ldots ,x_{t+r}){\text{ for }}{\mathcal {B}}(2r,2s+1,2t)} s s r + 1 ( x 1 , x 2 , … , x t + r + 1 ) s s r ( x 1 , x 2 , … , x t + r + 1 ) for B ( 2 r + 1 , 2 s , 2 t + 1 ) {\displaystyle s_{s^{r+1}}(x_{1},x_{2},\ldots ,x_{t+r+1})s_{s^{r}}(x_{1},x_{2},\ldots ,x_{t+r+1}){\text{ for }}{\mathcal {B}}(2r+1,2s,2t+1)} <p>supplies the desired <i>q</i>-analogue case. </p> <h3>Cyclically symmetric self-complementary plane partitions</h3> <p>A plane partition π {\displaystyle \pi } is called cyclically symmetric self-complementary if it is cyclically symmetric and self-complementary. The figure presents a cyclically symmetric self-complementary plane partition and the according matrix is below. </p> 4 4 4 1 3 3 2 1 3 2 1 1 3 {\displaystyle {\begin{matrix}4&4&4&1\\3&3&2&1\\3&2&1&1\\3&&&\end{matrix}}} <p>In a private communication with Stanley, Robbins conjectured that the total number of cyclically symmetric self-complementary plane partitions is given by N 6 ( 2 r , 2 r , 2 r ) {\displaystyle N_{6}(2r,2r,2r)} .<a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a><a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a> The total number of cyclically symmetric self-complementary plane partitions is given by </p> N 6 ( 2 r , 2 r , 2 r ) = D r 2 {\displaystyle N_{6}(2r,2r,2r)=D_{r}^{2}} <p> D r {\displaystyle D_{r}} is the number of r × r {\displaystyle r\times r} <a href="/facts/Alternating_sign_matrix/TMfXBu7l">alternating sign matrices</a>. A formula for D r {\displaystyle D_{r}} is given by </p> D r = ∏ j = 0 r − 1 ( 3 j + 1 ) ! ( r + j ) ! {\displaystyle D_{r}=\prod _{j=0}^{r-1}{\frac {(3j+1)!}{(r+j)!}}} <p><a href="/facts/Greg_Kuperberg/q64KLJNM">Greg Kuperberg</a> proved the formula for N 6 ( r , r , r ) {\displaystyle N_{6}(r,r,r)} in 1994.<a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a> </p> <h3>Totally symmetric self-complementary plane partitions</h3> <p>A totally symmetric self-complementary plane partition is a plane partition that is both totally symmetric and self-complementary. For instance, the matrix below is such a plane partition; it is visualised in the accompanying picture. </p> 6 6 6 5 5 3 6 5 5 3 3 1 6 5 5 3 3 1 5 3 3 1 1 5 3 3 1 1 3 1 1 {\displaystyle {\begin{matrix}6&6&6&5&5&3\\6&5&5&3&3&1\\6&5&5&3&3&1\\5&3&3&1&1&\\5&3&3&1&1&\\3&1&1&&&\end{matrix}}} <p>The formula N 7 ( r , r , r ) {\displaystyle N_{7}(r,r,r)} was conjectured by William H. Mills, Robbins and Howard Rumsey in their work <i>Self-Complementary Totally Symmetric Plane Partitions</i>.<a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a> The total number of totally symmetric self-complementary plane partitions is given by </p> N 7 ( 2 r , 2 r , 2 r ) = D r {\displaystyle N_{7}(2r,2r,2r)=D_{r}} <p>Andrews proves this formula in 1994 in his paper <i>Plane Partitions V: The TSSCPP Conjecture</i>.<a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Gaussian_binomial_coefficients/AWfKqp7F">Gaussian binomial coefficients</a></li> <li><a href="/facts/Voxel/xo077lPE">Voxel</a></li></ul> <ul><li><a href="/facts/George_Andrews_(mathematician)/x58GmSuR">G. Andrews</a>, <i>The Theory of Partitions</i>, Cambridge University Press, Cambridge, 1998, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-63766-X</li> <li>Bender, Edward A.; <a href="/facts/Donald_Knuth/GozNzDM6">Knuth, Donald E.</a> (1972), "Enumeration of plane partitions", <i>Journal of Combinatorial Theory, Series A</i>, 13: 40–54, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0097-3165%2872%2990007-6">10.1016/0097-3165(72)90007-6</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1096-0899">1096-0899</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0299574">0299574</a></li> <li><a href="/facts/Ian_G._Macdonald/PoaeNbeg">I.G. Macdonald</a>, <i>Symmetric Functions and Hall Polynomials</i>, Oxford University Press, Oxford, 1999, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-850450-0</li> <li><a href="/facts/Percy_Alexander_MacMahon/scUegUSn">P.A. MacMahon</a>, <i><a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABU9009">Combinatory analysis</a></i>, 2 vols, Cambridge University Press, 1915–16.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/PlanePartition.html">"Plane partition"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="http://dlmf.nist.gov/26.12">The DLMF page on Plane Partitions </a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Richard P. Stanley, Enumerative Combinatorics, Volume 2. Corollary 7.20.3. <a href="/wiki/Richard_P._Stanley" target="_blank">/wiki/Richard_P._Stanley</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>R.P. Stanley, Enumerative Combinatorics, Volume 2. pp. 365, 401–2. <a href="/wiki/Richard_P._Stanley" target="_blank">/wiki/Richard_P._Stanley</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>E. M. Wright, Asymptotic partition formulae I. Plane partitions, The Quarterly Journal of Mathematics 1 (1931) 177–189. <a href="/wiki/E._M._Wright" target="_blank">/wiki/E._M._Wright</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Here the typographical error (in Wright's paper) has been corrected, pointed out by Mutafchiev and Kamenov.[4] <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>MacMahon, Percy A. (1896). "XVI. Memoir on the theory of the partition of numbers.-Part I". Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 187: Article 52. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>MacMahon, Major Percy A. (1916). Combinatory Analysis Vol 2. Cambridge University Press. pp. §495. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>MacMahon, Major Percy A. (1916). Combinatory Analysis. Vol. 2. Cambridge University Press. pp. §429. <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>MacMahon, Major Percy A. (1916). Combinatory Analysis. Cambridge University Press. pp. §429, §494. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Kuperberg, Greg (1994). "Symmetries of plane partitions and the permanent-determinant method". Journal of Combinatorial Theory, Series A. 68: 115–151. arXiv:math/9410224. Bibcode:1994math.....10224K. doi:10.1016/0097-3165(94)90094-9. S2CID 14538036. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Macdonald, Ian G. (1998). Symmetric Functions and Hall Polynomials. Clarendon Press. pp. 20f, 85f. ISBN 9780198504504. <a href="9780198504504" target="_blank">9780198504504</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>MacMahon, Percy Alexander (1899). "Partitions of numbers whose graphs possess symmetry". Transactions of the Cambridge Philosophical Society. 17. <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Macdonald, Ian G. (1998). Symmetric Functions and Hall Polynomials. Clarendon Press. pp. 20f, 85f. ISBN 9780198504504. <a href="9780198504504" target="_blank">9780198504504</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Bender & Knuth (1972). "Enumeration of plane partitions". Journal of Combinatorial Theory, Series A. 13: 40–54. doi:10.1016/0097-3165(72)90007-6. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Andrews, George E. (1977). "Plane partitions II: The equivalence of the Bender-Knuth and MacMahon conjectures". Pacific Journal of Mathematics. 72 (2): 283–291. doi:10.2140/pjm.1977.72.283. <a href="https://doi.org/10.2140%2Fpjm.1977.72.283" target="_blank">https://doi.org/10.2140%2Fpjm.1977.72.283</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Andrews, George (1975). "Plane Partitions (I): The Mac Mahon Conjecture". Adv. Math. Suppl. Stud. 1. <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Macdonald, Ian G. (1998). Symmetric Functions and Hall Polynomials. Clarendon Press. pp. 83–86. ISBN 9780198504504. <a href="9780198504504" target="_blank">9780198504504</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Andrews, George E. (1977). "MacMahon's conjecture on symmetric plane partitions". Proceedings of the National Academy of Sciences. 74 (2): 426–429. Bibcode:1977PNAS...74..426A. doi:10.1073/pnas.74.2.426. PMC 392301. PMID 16592386. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC392301" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC392301</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Macdonald, Ian G. (1998). Symmetric Functions and Hall Polynomials. Clarendon Press. pp. 20f, 85f. ISBN 9780198504504. <a href="9780198504504" target="_blank">9780198504504</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Andrews, George E. (1979). "Plane Partitions(III): The Weak Macdonald Conjecture". Inventiones Mathematicae. 53 (3): 193–225. Bibcode:1979InMat..53..193A. doi:10.1007/bf01389763. S2CID 122528418. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Mills; Robbins; Rumsey (1982). "Proof of the Macdonald conjecture". Inventiones Mathematicae. 66: 73–88. Bibcode:1982InMat..66...73M. doi:10.1007/bf01404757. S2CID 120926391. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Stembridge, John R. (1995). "The Enumeration of Totally Symmetric Plane Partitions". Advances in Mathematics. 111 (2): 227–243. doi:10.1006/aima.1995.1023. <a href="https://doi.org/10.1006%2Faima.1995.1023" target="_blank">https://doi.org/10.1006%2Faima.1995.1023</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Andrews; Paule; Schneider (2005). "Plane Partitions VI: Stembridge's TSPP theorem". Advances in Applied Mathematics. 34 (4): 709–739. doi:10.1016/j.aam.2004.07.008. <a href="https://doi.org/10.1016%2Fj.aam.2004.07.008" target="_blank">https://doi.org/10.1016%2Fj.aam.2004.07.008</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>Bressoud, David M. (1999). Proofs and Confirmations. Cambridge University Press. pp. conj. 13. ISBN 9781316582756. <a href="9781316582756" target="_blank">9781316582756</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>Stanley, Richard P. (1970). "A Baker's dozen of conjectures concerning plane partitions". Combinatoire énumérative: 285–293. <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>Andrews, George (1980). "Totally symmetric plane partitions". Abstracts Amer. Math. Soc. 1: 415. <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>Koutschan; Kauers; Zeilberger (2011). "A proof of George Andrews' and David Robbins' q-TSPP conjecture". PNAS. 108 (6): 2196–2199. arXiv:1002.4384. Bibcode:2011PNAS..108.2196K. doi:10.1073/pnas.1019186108. PMC 3038772. S2CID 12077490. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3038772" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3038772</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> <li id="fn:27"><p>Stanley, Richard P. (1986). "Symmetries of Plane Partitions" (PDF). Journal of Combinatorial Theory, Series A. 43: 103–113. doi:10.1016/0097-3165(86)90028-2. <a href="http://www-math.mit.edu/~rstan/pubs/pubfiles/65.pdf" target="_blank">http://www-math.mit.edu/~rstan/pubs/pubfiles/65.pdf</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li> <li id="fn:28"><p>"Erratum". Journal of Combinatorial Theory. 43: 310. 1986. <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li> <li id="fn:29"><p>Stanley, Richard P. (1986). "Symmetries of Plane Partitions" (PDF). Journal of Combinatorial Theory, Series A. 43: 103–113. doi:10.1016/0097-3165(86)90028-2. <a href="http://www-math.mit.edu/~rstan/pubs/pubfiles/65.pdf" target="_blank">http://www-math.mit.edu/~rstan/pubs/pubfiles/65.pdf</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></p></li> <li id="fn:30"><p>Eisenkölbl, Theresia (2008). "A Schur function identity related to the (−1)-enumeration of self complementary plane partitions". Journal of Combinatorial Theory, Series A. 115 (2): 199–212. doi:10.1016/j.jcta.2007.05.004. <a href="https://doi.org/10.1016%2Fj.jcta.2007.05.004" target="_blank">https://doi.org/10.1016%2Fj.jcta.2007.05.004</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></p></li> <li id="fn:31"><p>Stanley, Richard P. (1971). "Theory and Application of Plane Partitions. Part 2". Studies in Applied Mathematics. 50 (3): 259–279. doi:10.1002/sapm1971503259. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></p></li> <li id="fn:32"><p>Stanley, Richard P. (1970). "A Baker's dozen of conjectures concerning plane partitions". Combinatoire énumérative: 285–293. <a href="#fnref:32" class="footnote-back-ref">↩</a></p></li> <li id="fn:33"><p>Stanley, Richard P. (1986). "Symmetries of Plane Partitions" (PDF). Journal of Combinatorial Theory, Series A. 43: 103–113. doi:10.1016/0097-3165(86)90028-2. <a href="http://www-math.mit.edu/~rstan/pubs/pubfiles/65.pdf" target="_blank">http://www-math.mit.edu/~rstan/pubs/pubfiles/65.pdf</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></p></li> <li id="fn:34"><p>Kuperberg, Greg (1994). "Symmetries of plane partitions and the permanent-determinant method". Journal of Combinatorial Theory, Series A. 68: 115–151. arXiv:math/9410224. Bibcode:1994math.....10224K. doi:10.1016/0097-3165(94)90094-9. S2CID 14538036. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></p></li> <li id="fn:35"><p>Mills; Robbins; Rumsey (1986). "Self-Complementary Totally Symmetric Plane Partitions". Journal of Combinatorial Theory, Series A. 42 (2): 277–292. doi:10.1016/0097-3165(86)90098-1. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></p></li> <li id="fn:36"><p>Andrews, George E. (1994). "Plane Partitions V: The TSSCPP Conjecture". Journal of Combinatorial Theory, Series A. 66: 28–39. doi:10.1016/0097-3165(94)90048-5. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></p></li> </ol>