CONGRUENCES MODULO 7 $7$ 7 FOR A CERTAIN SCHMIDT-TYPE PARTITION FUNCTION

Abstract Let S ( n ) $S(n)$ upper S left parenthesis n right parenthesis be the number of partitions a 1 + a 2 + a 3 + ⋯ $a_1+a_2+a_3+\\cdots $ a 1 plus a 2 plus a 3 plus midline horizontal ellipsis with a 1 ≥ a 2 ≥ a 3 ≥ ⋯ $a_1 \\geq a_2 \\geq a_3 \\geq \\cdots $ a 1 greater than or equals a 2 greater than or equals a 3 greater than or equals midline horizontal ellipsis such that n = a 1 + a 3 + a 5 + ⋯ $n = a_1+a_3+a_5+\\cdots $ n equals a 1 plus a 3 plus a 5 plus midline horizontal ellipsis and a 1 , a 3 , a 5 , … $a_1, a_3, a_5, \\ldots $ a 1 comma a 3 comma a 5 comma ellipsis are all even. We apply the action of Atkin’s U 7 $U_7$ upper U 7 operator to a certain