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The Qtspp Theorem The qTSPP Theorem Manuel Kauers RISC on a collaboration with Christoph Koutschan Doron Zeilberger and Tulane Rutgers The qTSPP Theorem Manuel Kauers RISC on a collaboration with Christoph Koutschan Doron Zeilberger and RISC Rutgers Richard Stanley Partitions n n A partition π of size n is a tuple (πi)i=1 ∈ N with n ≥ π1 ≥ π2 ≥···≥ πn. Partitions n n A partition π of size n is a tuple (πi)i=1 ∈ N with n ≥ π1 ≥ π2 ≥···≥ πn. Example: 5 3 3 2 1 0 is a partition of size 6 Partitions n n A partition π of size n is a tuple (πi)i=1 ∈ N with n ≥ π1 ≥ π2 ≥···≥ πn. Example: 5 3 3 2 1 0 is a partition of size 6 ttttt Picture: ttt ttt tt t Plane Partitions n n×n A plane partition π of size n is a matrix ((πi,j))i,j=1 ∈ N with n ≥ πi,1 ≥ πi,2 ≥···≥ πi,n and n ≥ π1,j ≥ π2,j ≥···≥ πn,j for all i and j. Plane Partitions n n×n A plane partition π of size n is a matrix ((πi,j))i,j=1 ∈ N with n ≥ πi,1 ≥ πi,2 ≥···≥ πi,n and n ≥ π1,j ≥ π2,j ≥···≥ πn,j for all i and j. 5 3 3 2 1 0 4 3 3 1 1 0 3 2 1 1 0 0 is a plane partition of size 6 2 2 1 1 0 0 2 1 0 0 0 0 1 1 0 0 0 0 Plane Partitions n n×n A plane partition π of size n is a matrix ((πi,j))i,j=1 ∈ N with n ≥ πi,1 ≥ πi,2 ≥···≥ πi,n and n ≥ π1,j ≥ π2,j ≥···≥ πn,j for all i and j. 5 3 3 2 1 0 4 3 3 1 1 0 3 2 1 1 0 0 2 2 1 1 0 0 2 1 0 0 0 0 1 1 0 0 0 0 Plane Partitions n n×n A plane partition π of size n is a matrix ((πi,j))i,j=1 ∈ N with n ≥ πi,1 ≥ πi,2 ≥···≥ πi,n and n ≥ π1,j ≥ π2,j ≥···≥ πn,j for all i and j. Plane Partitions n n×n A plane partition π of size n is a matrix ((πi,j))i,j=1 ∈ N with n ≥ πi,1 ≥ πi,2 ≥···≥ πi,n and n ≥ π1,j ≥ π2,j ≥···≥ πn,j for all i and j. Symmetric Plane Partitions n A symmetric plane partition π is a plane partition ((πi,j))i,j=1 ∈ n×n N with πi,j = πj,i for all i, j. Symmetric Plane Partitions n A symmetric plane partition π is a plane partition ((πi,j))i,j=1 ∈ n×n N with πi,j = πj,i for all i, j. Symmetric Plane Partitions n A symmetric plane partition π is a plane partition ((πi,j))i,j=1 ∈ n×n N with πi,j = πj,i for all i, j. Symmetric Plane Partitions n A symmetric plane partition π is a plane partition ((πi,j))i,j=1 ∈ n×n N with πi,j = πj,i for all i, j. Symmetric Plane Partitions n A symmetric plane partition π is a plane partition ((πi,j))i,j=1 ∈ n×n N with πi,j = πj,i for all i, j. Totally Symmetric Plane Partitions A totally symmetric plane partition π is a symmetric plane partition whose diagram is symmetric about all three diagonal planes. Totally Symmetric Plane Partitions A totally symmetric plane partition π is a symmetric plane partition whose diagram is symmetric about all three diagonal planes. Totally Symmetric Plane Partitions A totally symmetric plane partition π is a symmetric plane partition whose diagram is symmetric about all three diagonal planes. Totally Symmetric Plane Partitions A totally symmetric plane partition π is a symmetric plane partition whose diagram is symmetric about all three diagonal planes. Totally Symmetric Plane Partitions A totally symmetric plane partition π is a symmetric plane partition whose diagram is symmetric about all three diagonal planes. Totally Symmetric Plane Partitions Theorem: There are i + j + k − 1 i + j + k − 2 1≤i≤j≤k≤n totally symmetric plane partitions of size n. Totally Symmetric Plane Partitions Theorem: There are i + j + k − 1 i + j + k − 2 1≤i≤j≤k≤n totally symmetric plane partitions of size n. 1, 2, 5, 16, 66, 352, 2431, 21760, 252586, 3803648, 74327145, . Totally Symmetric Plane Partitions Theorem: There are i + j + k − 1 i + j + k − 2 1≤i≤j≤k≤n totally symmetric plane partitions of size n. 1, 2, 5, 16, 66, 352, 2431, 21760, 252586, 3803648, 74327145, . Totally Symmetric Plane Partitions Theorem: There are i + j + k − 1 i + j + k − 2 1≤i≤j≤k≤n totally symmetric plane partitions of size n. 1, 2, 5, 16, 66, 352, 2431, 21760, 252586, 3803648, 74327145, . Totally Symmetric Plane Partitions Theorem: There are i + j + k − 1 i + j + k − 2 1≤i≤j≤k≤n totally symmetric plane partitions of size n. 1, 2, 5, 16, 66, 352, 2431, 21760, 252586, 3803648, 74327145, . Totally Symmetric Plane Partitions Theorem: There are i + j + k − 1 i + j + k − 2 1≤i≤j≤k≤n totally symmetric plane partitions of size n. 1, 2, 5, 16, 66, 352, 2431, 21760, 252586, 3803648, 74327145, . Totally Symmetric Plane Partitions making 66 for n = 4. Totally Symmetric Plane Partitions First Proof: J. Stembridge, 1995. Totally Symmetric Plane Partitions First Proof: J. Stembridge, 1995. (without computer) Totally Symmetric Plane Partitions First Proof: J. Stembridge, 1995. (without computer) Second Proof: G. E. Andrews, P. Paule, C. Schneider, 2005. Totally Symmetric Plane Partitions First Proof: J. Stembridge, 1995. (without computer) Second Proof: G. E. Andrews, P. Paule, C. Schneider, 2005. Both proofs rely on Okada’s Lemma: Totally Symmetric Plane Partitions First Proof: J. Stembridge, 1995. (without computer) Second Proof: G. E. Andrews, P. Paule, C. Schneider, 2005. Both proofs rely on Okada’s Lemma: It is sufficient to show i + j + k − 1 2 det((a ))n = (n ≥ 1) i,j i,j=1 i + j + k − 2 1≤i≤j≤k≤n i+j−2 i+j−1 where ai,j = i−1 + i + 2δi,j − δi,j+1. a1,1 a1,2 a1,3 a1,4 a1,5 a1,6 a1,7 a1,8 · · · a2,1 a2,2 a2,3 a2,4 a2,5 a2,6 a2,7 a2,8 · · · a3,1 a3,2 a3,3 a3,4 a3,5 a3,6 a3,7 a3,8 · · · a4,1 a4,2 a4,3 a4,4 a4,5 a4,6 a4,7 a4,8 · · · a5,1 a5,2 a5,3 a5,4 a5,5 a5,6 a5,7 a5,8 · · · a6,1 a6,2 a6,3 a6,4 a6,5 a6,6 a6,7 a6,8 · · · a7,1 a7,2 a7,3 a7,4 a7,5 a7,6 a7,7 a7,8 · · · a8,1 a8,2 a8,3 a8,4 a8,5 a8,6 a8,7 a8,8 · · · a9,1 a9,2 a9,3 a9,4 a9,5 a9,6 a9,7 a9,8 · · · a10,1 a10,2 a10,3 a10,4 a10,5 a10,6 a10,7 a10,8 · · · a11,1 a11,2 a11,3 a11,4 a11,5 a11,6 a11,7 a11,8 · · · a12,1 a12,2 a12,3 a12,4 a12,5 a12,6 a12,7 a12,8 · · · . .. a1,1 a1,2 a1,3 a1,4 a1,5 a1,6 a1,7 a1,8 · · · a2,1 a2,2 a2,3 a2,4 a2,5 a2,6 a2,7 a2,8 · · · a3,1 a3,2 a3,3 a3,4 a3,5 a3,6 a3,7 a3,8 · · · a4,1 a4,2 a4,3 a4,4 a4,5 a4,6 a4,7 a4,8 · · · a5,1 a5,2 a5,3 a5,4 a5,5 a5,6 a5,7 a5,8 · · · a6,1 a6,2 a6,3 a6,4 a6,5 a6,6 a6,7 a6,8 · · · a7,1 a7,2 a7,3 a7,4 a7,5 a7,6 a7,7 a7,8 · · · a8,1 a8,2 a8,3 a8,4 a8,5 a8,6 a8,7 a8,8 · · · a9,1 a9,2 a9,3 a9,4 a9,5 a9,6 a9,7 a9,8 · · · a10,1 a10,2 a10,3 a10,4 a10,5 a10,6 a10,7 a10,8 · · · a11,1 a11,2 a11,3 a11,4 a11,5 a11,6 a11,7 a11,8 · · · a12,1 a12,2 a12,3 a12,4 a12,5 a12,6 a12,7 a12,8 · · · . .. a1,1 a1,2 a1,3 a1,4 a1,5 a1,6 a1,7 a1,8 · · · a2,1 a2,2 a2,3 a2,4 a2,5 a2,6 a2,7 a2,8 · · · a3,1 a3,2 a3,3 a3,4 a3,5 a3,6 a3,7 a3,8 · · · a4,1 a4,2 a4,3 a4,4 a4,5 a4,6 a4,7 a4,8 · · · a5,1 a5,2 a5,3 a5,4 a5,5 a5,6 a5,7 a5,8 · · · a6,1 a6,2 a6,3 a6,4 a6,5 a6,6 a6,7 a6,8 · · · a7,1 a7,2 a7,3 a7,4 a7,5 a7,6 a7,7 a7,8 · · · a8,1 a8,2 a8,3 a8,4 a8,5 a8,6 a8,7 a8,8 · · · a9,1 a9,2 a9,3 a9,4 a9,5 a9,6 a9,7 a9,8 · · · a10,1 a10,2 a10,3 a10,4 a10,5 a10,6 a10,7 a10,8 · · · a11,1 a11,2 a11,3 a11,4 a11,5 a11,6 a11,7 a11,8 · · · a12,1 a12,2 a12,3 a12,4 a12,5 a12,6 a12,7 a12,8 · · · .
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