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History of Schur polynomials $s_\lambda$

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2026-09-16 11:23 bmatschke interactive tags: bases for symmetric polynomials. a tag is a family of tables, and these are the members of one. numberdb-data#92 asked for it; the membership is decided by the generating function each table gives, not by its title. current reviewed
2026-09-12 15:25 bmatschke attached generate.py
2026-09-01 22:15 bmatschke attach each link to the name it follows, so the name is the link
2026-08-27 11:06 bmatschke generator the Schur polynomials, by Jacobi-Trudi
2026-08-27 11:06 bmatschke with claude-opus-5 interactive proposed from schur.yaml

What changed between 2026-09-12 15:25 and 2026-09-16 11:23

from line 46 (5 lines, 1 more than before) @@ -46,4 +46,5 @@ 
 - combinatorics - algebra+- bases for symmetric polynomials Programs:   program-sage:
from line 57 (59 lines, 95 fewer than before) @@ -56,154 +57,59 @@ 
       \ # a partition of 5, past this table" Numbers:-- params:-    n: '1'-    lambda: '1'-  number: x1-- params:-    n: '1'-    lambda: '2'-  number: x1^2-- params:-    n: '1'-    lambda: '3'-  number: x1^3-- params:-    n: '1'-    lambda: '4'-  number: x1^4-- params:-    n: '2'-    lambda: '1'-  number: x1 + x2-- params:-    n: '2'-    lambda: '2'-  number: x1^2 + x1*x2 + x2^2-- params:-    n: '2'-    lambda: 1,1-  number: x1*x2-- params:-    n: '2'-    lambda: '3'-  number: x1^3 + x1^2*x2 + x1*x2^2 + x2^3-- params:-    n: '2'-    lambda: 2,1-  number: x1^2*x2 + x1*x2^2-- params:-    n: '2'-    lambda: '4'-  number: x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4-- params:-    n: '2'-    lambda: 3,1-  number: x1^3*x2 + x1^2*x2^2 + x1*x2^3-- params:-    n: '2'-    lambda: 2,2-  number: x1^2*x2^2-- params:-    n: '3'-    lambda: '1'-  number: x1 + x2 + x3-- params:-    n: '3'-    lambda: '2'-  number: x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2-- params:-    n: '3'-    lambda: 1,1-  number: x1*x2 + x1*x3 + x2*x3-- params:-    n: '3'-    lambda: '3'-  number: x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2-    + x2*x3^2 + x3^3-- params:-    n: '3'-    lambda: 2,1-  number: x1^2*x2 + x1*x2^2 + x1^2*x3 + 2*x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2-- params:-    n: '3'-    lambda: 1,1,1-  number: x1*x2*x3-- params:-    n: '3'-    lambda: '4'-  number: x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4 + x1^3*x3 + x1^2*x2*x3 + x1*x2^2*x3-    + x2^3*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3 + x3^4-- params:-    n: '3'-    lambda: 3,1-  number: x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x1^3*x3 + 2*x1^2*x2*x3 + 2*x1*x2^2*x3 +-    x2^3*x3 + x1^2*x3^2 + 2*x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3-- params:-    n: '3'-    lambda: 2,2-  number: x1^2*x2^2 + x1^2*x2*x3 + x1*x2^2*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2-- params:-    n: '3'-    lambda: 2,1,1-  number: x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2-- params:-    n: '4'-    lambda: '1'-  number: x1 + x2 + x3 + x4-- params:-    n: '4'-    lambda: '2'-  number: x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1*x4 + x2*x4 + x3*x4 + x4^2-- params:-    n: '4'-    lambda: 1,1-  number: x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4-- params:-    n: '4'-    lambda: '3'-  number: x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2-    + x2*x3^2 + x3^3 + x1^2*x4 + x1*x2*x4 + x2^2*x4 + x1*x3*x4 + x2*x3*x4 + x3^2*x4-    + x1*x4^2 + x2*x4^2 + x3*x4^2 + x4^3-- params:-    n: '4'-    lambda: 2,1-  number: x1^2*x2 + x1*x2^2 + x1^2*x3 + 2*x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 +-    x1^2*x4 + 2*x1*x2*x4 + x2^2*x4 + 2*x1*x3*x4 + 2*x2*x3*x4 + x3^2*x4 + x1*x4^2 +-    x2*x4^2 + x3*x4^2-- params:-    n: '4'-    lambda: 1,1,1-  number: x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4-- params:-    n: '4'-    lambda: '4'-  number: x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4 + x1^3*x3 + x1^2*x2*x3 + x1*x2^2*x3-    + x2^3*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3 + x3^4 + x1^3*x4-    + x1^2*x2*x4 + x1*x2^2*x4 + x2^3*x4 + x1^2*x3*x4 + x1*x2*x3*x4 + x2^2*x3*x4 +-    x1*x3^2*x4 + x2*x3^2*x4 + x3^3*x4 + x1^2*x4^2 + x1*x2*x4^2 + x2^2*x4^2 + x1*x3*x4^2-    + x2*x3*x4^2 + x3^2*x4^2 + x1*x4^3 + x2*x4^3 + x3*x4^3 + x4^4-- params:-    n: '4'-    lambda: 3,1-  number: x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x1^3*x3 + 2*x1^2*x2*x3 + 2*x1*x2^2*x3 +-    x2^3*x3 + x1^2*x3^2 + 2*x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3 + x1^3*x4 +-    2*x1^2*x2*x4 + 2*x1*x2^2*x4 + x2^3*x4 + 2*x1^2*x3*x4 + 3*x1*x2*x3*x4 + 2*x2^2*x3*x4-    + 2*x1*x3^2*x4 + 2*x2*x3^2*x4 + x3^3*x4 + x1^2*x4^2 + 2*x1*x2*x4^2 + x2^2*x4^2-    + 2*x1*x3*x4^2 + 2*x2*x3*x4^2 + x3^2*x4^2 + x1*x4^3 + x2*x4^3 + x3*x4^3-- params:-    n: '4'-    lambda: 2,2-  number: x1^2*x2^2 + x1^2*x2*x3 + x1*x2^2*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2-    + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4 + 2*x1*x2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4-    + x2*x3^2*x4 + x1^2*x4^2 + x1*x2*x4^2 + x2^2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2 +-    x3^2*x4^2-- params:-    n: '4'-    lambda: 2,1,1-  number: x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2 + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4-    + 3*x1*x2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x1*x2*x4^2 + x1*x3*x4^2-    + x2*x3*x4^2-- params:-    n: '4'-    lambda: 1,1,1,1-  number: x1*x2*x3*x4+  '1':+    '1': x1+    '2': x1^2+    '3': x1^3+    '4': x1^4+  '2':+    '1': x1 + x2+    '2': x1^2 + x1*x2 + x2^2+    1,1: x1*x2+    '3': x1^3 + x1^2*x2 + x1*x2^2 + x2^3+    2,1: x1^2*x2 + x1*x2^2+    '4': x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4+    3,1: x1^3*x2 + x1^2*x2^2 + x1*x2^3+    2,2: x1^2*x2^2+  '3':+    '1': x1 + x2 + x3+    '2': x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2+    1,1: x1*x2 + x1*x3 + x2*x3+    '3': x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2+      + x2*x3^2 + x3^3+    2,1: x1^2*x2 + x1*x2^2 + x1^2*x3 + 2*x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2+    1,1,1: x1*x2*x3+    '4': x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4 + x1^3*x3 + x1^2*x2*x3 + x1*x2^2*x3+      + x2^3*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3 + x3^4+    3,1: x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x1^3*x3 + 2*x1^2*x2*x3 + 2*x1*x2^2*x3 + x2^3*x3+      + x1^2*x3^2 + 2*x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3+    2,2: x1^2*x2^2 + x1^2*x2*x3 + x1*x2^2*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2+    2,1,1: x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2+  '4':+    '1': x1 + x2 + x3 + x4+    '2': x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1*x4 + x2*x4 + x3*x4 + x4^2+    1,1: x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4+    '3': x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2+      + x2*x3^2 + x3^3 + x1^2*x4 + x1*x2*x4 + x2^2*x4 + x1*x3*x4 + x2*x3*x4 + x3^2*x4+      + x1*x4^2 + x2*x4^2 + x3*x4^2 + x4^3+    2,1: x1^2*x2 + x1*x2^2 + x1^2*x3 + 2*x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 ++      x1^2*x4 + 2*x1*x2*x4 + x2^2*x4 + 2*x1*x3*x4 + 2*x2*x3*x4 + x3^2*x4 + x1*x4^2+      + x2*x4^2 + x3*x4^2+    1,1,1: x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4+    '4': x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4 + x1^3*x3 + x1^2*x2*x3 + x1*x2^2*x3+      + x2^3*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3 + x3^4 ++      x1^3*x4 + x1^2*x2*x4 + x1*x2^2*x4 + x2^3*x4 + x1^2*x3*x4 + x1*x2*x3*x4 + x2^2*x3*x4+      + x1*x3^2*x4 + x2*x3^2*x4 + x3^3*x4 + x1^2*x4^2 + x1*x2*x4^2 + x2^2*x4^2 + x1*x3*x4^2+      + x2*x3*x4^2 + x3^2*x4^2 + x1*x4^3 + x2*x4^3 + x3*x4^3 + x4^4+    3,1: x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x1^3*x3 + 2*x1^2*x2*x3 + 2*x1*x2^2*x3 + x2^3*x3+      + x1^2*x3^2 + 2*x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3 + x1^3*x4 + 2*x1^2*x2*x4+      + 2*x1*x2^2*x4 + x2^3*x4 + 2*x1^2*x3*x4 + 3*x1*x2*x3*x4 + 2*x2^2*x3*x4 + 2*x1*x3^2*x4+      + 2*x2*x3^2*x4 + x3^3*x4 + x1^2*x4^2 + 2*x1*x2*x4^2 + x2^2*x4^2 + 2*x1*x3*x4^2+      + 2*x2*x3*x4^2 + x3^2*x4^2 + x1*x4^3 + x2*x4^3 + x3*x4^3+    2,2: x1^2*x2^2 + x1^2*x2*x3 + x1*x2^2*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2+      + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4 + 2*x1*x2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4+      + x2*x3^2*x4 + x1^2*x4^2 + x1*x2*x4^2 + x2^2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2+      + x3^2*x4^2+    2,1,1: x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2 + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4+      + 3*x1*x2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x1*x2*x4^2 + x1*x3*x4^2+      + x2*x3*x4^2+    1,1,1,1: x1*x2*x3*x4

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