History of Elementary symmetric polynomials $e_k$

 - combinatorics - algebra+- bases for symmetric polynomials Programs:   program-sage:

       \ # more variables than this table holds" Numbers:-- params:-    n: '1'-    k: '1'-  number: x1-- params:-    n: '2'-    k: '1'-  number: x1 + x2-- params:-    n: '2'-    k: '2'-  number: x1*x2-- params:-    n: '3'-    k: '1'-  number: x1 + x2 + x3-- params:-    n: '3'-    k: '2'-  number: x1*x2 + x1*x3 + x2*x3-- params:-    n: '3'-    k: '3'-  number: x1*x2*x3-- params:-    n: '4'-    k: '1'-  number: x1 + x2 + x3 + x4-- params:-    n: '4'-    k: '2'-  number: x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4-- params:-    n: '4'-    k: '3'-  number: x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4-- params:-    n: '4'-    k: '4'-  number: x1*x2*x3*x4-- params:-    n: '5'-    k: '1'-  number: x1 + x2 + x3 + x4 + x5-- params:-    n: '5'-    k: '2'-  number: x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4 + x1*x5 + x2*x5 + x3*x5 +-    x4*x5-- params:-    n: '5'-    k: '3'-  number: x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4 + x1*x2*x5 + x1*x3*x5 + x2*x3*x5-    + x1*x4*x5 + x2*x4*x5 + x3*x4*x5-- params:-    n: '5'-    k: '4'-  number: x1*x2*x3*x4 + x1*x2*x3*x5 + x1*x2*x4*x5 + x1*x3*x4*x5 + x2*x3*x4*x5-- params:-    n: '5'-    k: '5'-  number: x1*x2*x3*x4*x5-- params:-    n: '6'-    k: '1'-  number: x1 + x2 + x3 + x4 + x5 + x6-- params:-    n: '6'-    k: '2'-  number: x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4 + x1*x5 + x2*x5 + x3*x5 +-    x4*x5 + x1*x6 + x2*x6 + x3*x6 + x4*x6 + x5*x6-- params:-    n: '6'-    k: '3'-  number: x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4 + x1*x2*x5 + x1*x3*x5 + x2*x3*x5-    + x1*x4*x5 + x2*x4*x5 + x3*x4*x5 + x1*x2*x6 + x1*x3*x6 + x2*x3*x6 + x1*x4*x6 +-    x2*x4*x6 + x3*x4*x6 + x1*x5*x6 + x2*x5*x6 + x3*x5*x6 + x4*x5*x6-- params:-    n: '6'-    k: '4'-  number: x1*x2*x3*x4 + x1*x2*x3*x5 + x1*x2*x4*x5 + x1*x3*x4*x5 + x2*x3*x4*x5 + x1*x2*x3*x6-    + x1*x2*x4*x6 + x1*x3*x4*x6 + x2*x3*x4*x6 + x1*x2*x5*x6 + x1*x3*x5*x6 + x2*x3*x5*x6-    + x1*x4*x5*x6 + x2*x4*x5*x6 + x3*x4*x5*x6-- params:-    n: '6'-    k: '5'-  number: x1*x2*x3*x4*x5 + x1*x2*x3*x4*x6 + x1*x2*x3*x5*x6 + x1*x2*x4*x5*x6 + x1*x3*x4*x5*x6-    + x2*x3*x4*x5*x6-- params:-    n: '6'-    k: '6'-  number: x1*x2*x3*x4*x5*x6+  '1':+    '1': x1+  '2':+    '1': x1 + x2+    '2': x1*x2+  '3':+    '1': x1 + x2 + x3+    '2': x1*x2 + x1*x3 + x2*x3+    '3': x1*x2*x3+  '4':+    '1': x1 + x2 + x3 + x4+    '2': x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4+    '3': x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4+    '4': x1*x2*x3*x4+  '5':+    '1': x1 + x2 + x3 + x4 + x5+    '2': x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4 + x1*x5 + x2*x5 + x3*x5 + x4*x5+    '3': x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4 + x1*x2*x5 + x1*x3*x5 + x2*x3*x5+      + x1*x4*x5 + x2*x4*x5 + x3*x4*x5+    '4': x1*x2*x3*x4 + x1*x2*x3*x5 + x1*x2*x4*x5 + x1*x3*x4*x5 + x2*x3*x4*x5+    '5': x1*x2*x3*x4*x5+  '6':+    '1': x1 + x2 + x3 + x4 + x5 + x6+    '2': x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4 + x1*x5 + x2*x5 + x3*x5 + x4*x5+      + x1*x6 + x2*x6 + x3*x6 + x4*x6 + x5*x6+    '3': x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4 + x1*x2*x5 + x1*x3*x5 + x2*x3*x5+      + x1*x4*x5 + x2*x4*x5 + x3*x4*x5 + x1*x2*x6 + x1*x3*x6 + x2*x3*x6 + x1*x4*x6+      + x2*x4*x6 + x3*x4*x6 + x1*x5*x6 + x2*x5*x6 + x3*x5*x6 + x4*x5*x6+    '4': x1*x2*x3*x4 + x1*x2*x3*x5 + x1*x2*x4*x5 + x1*x3*x4*x5 + x2*x3*x4*x5 + x1*x2*x3*x6+      + x1*x2*x4*x6 + x1*x3*x4*x6 + x2*x3*x4*x6 + x1*x2*x5*x6 + x1*x3*x5*x6 + x2*x3*x5*x6+      + x1*x4*x5*x6 + x2*x4*x5*x6 + x3*x4*x5*x6+    '5': x1*x2*x3*x4*x5 + x1*x2*x3*x4*x6 + x1*x2*x3*x5*x6 + x1*x2*x4*x5*x6 + x1*x3*x4*x5*x6+      + x2*x3*x4*x5*x6+    '6': x1*x2*x3*x4*x5*x6