Binomial Determinants for Tiling Problems Yield to the Holonomic Ansatz Hao Du, Christoph Koutschan, Thotsaporn Thanatipanonda, Elaine Wong Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences July 2, 2021 S´eminairede Combinatoire de Lyon `al'ENS + 1 + 1 + 1 µ µ + i + j + s + t − 4 Es;t(n) := det − δi+s;j+t : 16i6n j + t − 1 16j6n Example: is the determinant of the matrix 0 µ+8 µ+9 µ+10 µ+11 µ+12 1 B C B µ+9 µ+10 µ+11 µ+12 µ+13 C B C B C B µ+10 µ+11 µ+12 µ+13 µ+14 C B C B µ+11 µ+12 µ+13 µ+14 µ+15 C B C @ A µ+12 µ+13 µ+14 µ+15 µ+16 Families of Binomial Determinants Definition: For n 2 N, for s; t 2 Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds;t(n) := det + δi+s;j+t ; 16i6n j + t − 1 16j6n 1 / 32 + 1 + 1 + 1 Example: is the determinant of the matrix 0 µ+8 µ+9 µ+10 µ+11 µ+12 1 B C B µ+9 µ+10 µ+11 µ+12 µ+13 C B C B C B µ+10 µ+11 µ+12 µ+13 µ+14 C B C B µ+11 µ+12 µ+13 µ+14 µ+15 C B C @ A µ+12 µ+13 µ+14 µ+15 µ+16 Families of Binomial Determinants Definition: For n 2 N, for s; t 2 Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds;t(n) := det + δi+s;j+t ; 16i6n j + t − 1 16j6n µ µ + i + j + s + t − 4 Es;t(n) := det − δi+s;j+t : 16i6n j + t − 1 16j6n 1 / 32 + 1 + 1 + 1 Example: is the determinant of the matrix 0 µ+8 µ+9 µ+10 µ+11 µ+12 1 B C B µ+9 µ+10 µ+11 µ+12 µ+13 C B C B C B µ+10 µ+11 µ+12 µ+13 µ+14 C B C B µ+11 µ+12 µ+13 µ+14 µ+15 C B C @ A µ+12 µ+13 µ+14 µ+15 µ+16 Families of Binomial Determinants Definition: For n 2 N, for s; t 2 Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds;t(n) := det + δi+s;j+t ; 16i6n j + t − 1 16j6n µ µ + i + j + s + t − 4 Es;t(n) := det − δi+s;j+t : 16i6n j + t − 1 16j6n µ History: D0;0(n) was introduced in the work of Andrews in 1979 { 1980 in the context of descending plane partitions: 1 / 32 Families of Binomial Determinants Definition: For n 2 N, for s; t 2 Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds;t(n) := det + δi+s;j+t ; 16i6n j + t − 1 16j6n µ µ + i + j + s + t − 4 Es;t(n) := det − δi+s;j+t : 16i6n j + t − 1 16j6n µ Example: D4;6(5) is the determinant of the matrix 0 µ+8 µ+9 µ+10 µ+11 µ+12 1 6 7 8 9 10 B C B µ+9 µ+10 µ+11 µ+12 µ+13 C B 6 7 8 9 10 C B C B µ+10 + 1 µ+11 µ+12 µ+13 µ+14 C B 6 7 8 9 10 C B µ+11 µ+12 µ+13 µ+14 µ+15 C B + 1 C @ 6 7 8 9 10 A µ+12 µ+13 µ+14 µ+15 µ+16 6 7 8 + 1 9 10 1 / 32 Families of Binomial Determinants Definition: For n 2 N, for s; t 2 Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds;t(n) := det + δi+s;j+t ; 16i6n j + t − 1 16j6n µ µ + i + j + s + t − 4 Es;t(n) := det − δi+s;j+t : 16i6n j + t − 1 16j6n µ+2 Example: D3;5 (5) is the determinant of the matrix 0 µ+8 µ+9 µ+10 µ+11 µ+12 1 5 6 7 8 9 B C B µ+9 µ+10 µ+11 µ+12 µ+13 C B 5 6 7 8 9 C B C B µ+10 + 1 µ+11 µ+12 µ+13 µ+14 C B 5 6 7 8 9 C B µ+11 µ+12 µ+13 µ+14 µ+15 C B + 1 C @ 5 6 7 8 9 A µ+12 µ+13 µ+14 µ+15 µ+16 5 6 7 + 1 8 9 1 / 32 Families of Binomial Determinants Definition: For n 2 N, for s; t 2 Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds;t(n) := det + δi+s;j+t ; 16i6n j + t − 1 16j6n µ µ + i + j + s + t − 4 Es;t(n) := det − δi+s;j+t : 16i6n j + t − 1 16j6n µ Example: D4;6(5) is the determinant of the matrix 0 µ+8 µ+9 µ+10 µ+11 µ+12 1 6 7 8 9 10 B C B µ+9 µ+10 µ+11 µ+12 µ+13 C B 6 7 8 9 10 C B C B µ+10 + 1 µ+11 µ+12 µ+13 µ+14 C B 6 7 8 9 10 C B µ+11 µ+12 µ+13 µ+14 µ+15 C B + 1 C @ 6 7 8 9 10 A µ+12 µ+13 µ+14 µ+15 µ+16 6 7 8 + 1 9 10 1 / 32 + 1 + 1 + 1 Families of Binomial Determinants Definition: For n 2 N, for s; t 2 Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds;t(n) := det + δi+s;j+t ; 16i6n j + t − 1 16j6n µ µ + i + j + s + t − 4 Es;t(n) := det − δi+s;j+t : 16i6n j + t − 1 16j6n µ−1 Example: D5;6 (5) is the determinant of the matrix 0 µ+8 µ+9 µ+10 µ+11 µ+12 1 6 7 8 9 10 B C B µ+9 + 1 µ+10 µ+11 µ+12 µ+13 C B 6 7 8 9 10 C B C B µ+10 µ+11 + 1 µ+12 µ+13 µ+14 C B 6 7 8 9 10 C B µ+11 µ+12 µ+13 µ+14 µ+15 C B + 1 C @ 6 7 8 9 10 A µ+12 µ+13 µ+14 µ+15 µ+16 6 7 8 9 + 1 10 1 / 32 + 1 + 1 + 1 Families of Binomial Determinants Definition: For n 2 N, for s; t 2 Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds;t(n) := det + δi+s;j+t ; 16i6n j + t − 1 16j6n µ µ + i + j + s + t − 4 Es;t(n) := det − δi+s;j+t : 16i6n j + t − 1 16j6n µ−1 Example: E5;6 (5) is the determinant of the matrix 0 µ+8 µ+9 µ+10 µ+11 µ+12 1 6 7 8 9 10 B C B µ+9 − 1 µ+10 µ+11 µ+12 µ+13 C B 6 7 8 9 10 C B C B µ+10 µ+11 − 1 µ+12 µ+13 µ+14 C B 6 7 8 9 10 C B µ+11 µ+12 µ+13 µ+14 µ+15 C B − 1 C @ 6 7 8 9 10 A µ+12 µ+13 µ+14 µ+15 µ+16 6 7 8 9 − 1 10 1 / 32 \Conjecture 37" (Lascoux/Krattenthaler 2005) Let µ be an indeterminate and m; r 2 Z. If m > r > 1, then µ m−r 4m+(m−r)(m−r−1)−3r E1;2r−1(2m − 1) = (−1) · 2 2r−3 ! m−1 ! Y Y i!(i + 1)! × i! · (2i)! (2i + 2)! i=0 i=0 2r−2 ! µ 1 Y × (µ − 1) · 2 + r − 2 m−r · (µ + i − 1)2m+2r−2i−1 i=1 r−2 2 ! Y (2m − 2i − 3)! × 2 i=0 (m − i − 2)! (2m + 2i − 1)! (2m + 2i + 1)! 0 m−r−1 1 b 2 c Y 2 2 × B µ + 3i + 3r − 1 − µ − 3m + 3i + 3 C @ 2 2 m−r−2i−1 2 m−r−2iA i=0 2 / 32 µ µ Es;t(n) Family Ds;t(n) Family t t n odd n even s s t t n even n odd s s 3 / 32 µ µ Es;t(n) Family Ds;t(n) Family t t n odd n even s s t t n even n odd s s 3 / 32 µ µ Es;t(n) Family Ds;t(n) Family t t n odd n even s s t t n even n odd s s 3 / 32 µ µ Es;t(n) Family Ds;t(n) Family t t n odd n even s s t t n even n odd s s 3 / 32 µ µ Es;t(n) Family Ds;t(n) Family t t n odd n even s s t t n even n odd s s 3 / 32 µ µ Es;t(n) Family Ds;t(n) Family t t n odd n even s s t t n even n odd s s 3 / 32 µ µ Es;t(n) Family Ds;t(n) Family t t n odd n even s s t t n even n odd s s 3 / 32 µ µ Es;t(n) Family Ds;t(n) Family t t n odd n even s s t t n even n odd s s 3 / 32 µ µ Es;t(n) Family Ds;t(n) Family t t n odd n even s s t t n even n odd s s 3 / 32 µ µ Es;t(n) Family Ds;t(n) Family t t n odd n even s s t t n even n odd s s 3 / 32 µ µ Es;t(n) Family Ds;t(n) Family t t n odd n even s s t t n even n odd s s 3 / 32 µ µ Es;t(n) Family Ds;t(n) Family t t n odd n even s s t t n even n odd s s 3 / 32 µ µ Es;t(n) Family Ds;t(n) Family t t n odd n even s s t t n even n odd s s 3 / 32 µ µ Es;t(n) Family Ds;t(n) Family t t n odd n even s s t t n even n odd s s 3 / 32 µ µ Es;t(n) Family Ds;t(n) Family t t n odd n even s s t t n even n odd s s 3 / 32 For each path P , let !(P ) be the product of its edge weights.